proposed form for the atmospheric turbulence spatial spectrum at large scales

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D. P. Greenwood and D. O. Tarazano Vol. 25, No. 6 /June 2008/J. Opt. Soc. Am. A 1349

Proposed form for the atmospheric turbulencespatial spectrum at large scales

Darryl P. Greenwood1,* and Donald O. Tarazano2

1Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, Massachusetts 02420, USA2161 Terrace Villa Drive, Centerville, Ohio 45459, USA

*Corresponding author: [email protected]

Received February 12, 2008; revised April 6, 2008; accepted April 7, 2008;posted April 9, 2008 (Doc. ID 92673); published May 16, 2008

The performance of large optical systems can be strongly influenced by the behavior of large-scale atmosphericturbulence. Previous optical phase difference data diverged from theoretical predictions based on vonKármán’s model. To establish a better model microthermal data over large spatial scales were collected at twosites with few terrain inhomogeneities, in unstable midday conditions. Spatial structure function and spectraare developed. A model spatial spectrum is fit to the data with parameters to set two power law dependenciesand the transition rate between them. By working with spatial spectral data the assumption of frozen flow isavoided. The optimum spatial spectrum is Kolmogorov-like in the inertial range. Temporal spectral data com-pare favorably with the new model, which better describes the transition of turbulence scaling from smallscales to large.

OCIS codes: 010.7060, 010.1330, 010.1300, 010.3920, 350.1270.

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. INTRODUCTIONearly all optical atmospheric propagation theory derives

rom the spatiotemporal spectrum of microtemperature,s temperature turbulence almost exclusively governs re-ractive index fluctuations in the visible to infrared wave-engths. If the theoretical spectrum was incorrect theheories would require modification. As early as 1972, dis-repancies were reported by Huber and Urtz [1] in com-aring optical phase difference spectra with theory de-ived from the well-known von Kármán model. Thisrompted a fundamental investigation of the low fre-uency behavior of microtemperature fluctuations and,ubsequently, the development of a model that better fitshe data. It is the development of the revised model basedn spatial and temporal microtemperature measurementshat is the subject of this paper, with emphasis on the lowavenumber, input range of turbulence.Research reported in this paper was conducted in the

arly 1970s at a U.S. Air Force Laboratory [the Rome Airevelopment Center (RADC)] and reported in an in-house

echnical report [2]. Though never published formally in aefereed journal until now, the work has found circulationargely due to informal communications and now throughhe Internet. The work is perhaps more timely than ever,ith increasing emphasis on large optical systems includ-

ng wide-baseline interferometry and huge ground-basedelescopes. References to this work in this context can beound in Gardner [3] and Buscher et al. [4], though nei-her author could come to a firm conclusion on the valid-ty of the proposed 1974 model. In 1995, Voitsekhovich [5]ompared three models, including the one developed here,nd concluded that there was not much difference for the-retical analyses of Zernike coefficients for aperture sizesmaller than the outer scale. In 2001 Wheelon [6] consid-red a number of large- and small-scale turbulence mod-

1084-7529/08/061349-12/$15.00 © 2

ls and their applicability to atmospheric propagationodels, citing the model developed here. Then, in 2007,heelon et al. [7] published what may be the most defini-

ive and highest quality corroborative work, thus prompt-ng this submission. Though late publication such as thateported here may be unusual, the material presentedere appears to be the earliest definitive work in thisrea.The first indication there was a problem with the the-

retical model was a measurement of the spatial spec-rum of optical phase difference power spectrum of apherical wave at 10.6 �m wavelength [1]. The data dem-nstrated more power in the low frequencies than wasredicted by the theory of Reinhardt and Collins [8]. Evenhen the theory was modified to account for a fluctuatingind direction, the agreement was still not good. This leds to suspect the bases for the theory: 1. the Rytov ap-roximation in the solution of the wave equation, 2. thepatial refractive index spectrum, and 3. the frozen-ow—or Taylor—hypothesis. The Rytov approximationhould hold equally well in the low and high frequenciesnd thus should not be suspect in the calculations ofhase quantities. Since the microtemperature dominatesefractive index fluctuations, we conducted a series of ex-eriments to verify the von Kármán spatial spectrum andhe frozen-flow hypothesis for low frequencies.

The input range of the spatial spectrum is governed byhe mechanisms of turbulence generation and a beginningascade process. Various researchers [9] have reportedhat the atmosphere imposes several boundary conditionsimultaneously, each with differing outer scale lengths. Inonvective daytime conditions (reported here), at leastwo large-scale dimensions—the height above ground andhe height of the inversion layer—have a strong influencen both wind and temperature spectra at frequencies be-

008 Optical Society of America

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1350 J. Opt. Soc. Am. A/Vol. 25, No. 6 /June 2008 D. P. Greenwood and D. O. Tarazano

ow the inertial range. Kaimal et al. [10] show how spec-ra transition from high to low frequency and how theransition changes with the height above ground and theeight of the boundary layer. We will see that for mea-urement heights of 2 to 33 m (i.e., much less than a day-ime inversion), outer scale sizes are, rather consistently,ypically �2 times the height above ground. This is ingreement with Kaimal’s statement [10] that for heightsess than 0.1 times the inversion height, the controllingcale is height above ground.

In addition to the outer scale becoming of increasingmportance to large optical systems, the outer scale andhe detailed spectrum of turbulence are important to un-erstanding refractive index fluctuations in general asell as to broader questions in meterological modeling.yngaard et al. [11] addresses numerical modeling of

lectromagnetic propagation in the atmosphere, which re-uires turbulence models in this spectral domain. In 2001akanishi [12] addressed large-eddy simulation and pro-osed updated mesoscale meteorological models that needalues for the outer scale (the “master scale”) and aeans of accurately describing the large-scale spectra.Since the spatial structure function of temperature is

elated through a Fourier transform to the spatial spec-rum and does not depend on the frozen-flow hypothesis,e use the structure function as a check on the validity of

he von Kármán spectrum. A von Kármán spectrum isound to be a less than ideal fit model to the data, so anmpirical spectrum is developed. The model is then inte-rated to give the temporal spectrum, which in turn isompared with data. At that point the residual error cannly be attributed to the frozen-flow hypothesis. We willee that the hypothesis works remarkably well in the low,nput range frequencies. Although a specific spectrum isroposed the mathematics will be kept sufficiently gen-ral so that it can be extended to analyses of non-olmogorov turbulence or to different wavenumberependencies.

. THEORY. Classical Spectrae begin by briefly reviewing the history of the vonármán spectrum and the frozen-flow hypothesis. Theost basic form of the temperature spatial spectrum was

eveloped in work by Kolmogorov [13,14] and via an in-ependent method by Obukhov [15] and applies only tohe inertial subrange, �0

−1��0−1,

�T�� −11/3. �1�

cale lengths �0 and �0 define the limits of the inertialubrange and are the internal and external scales ofanchev [16]. Tatarskii [17] uses the symbols l0 and L0,espectively, as the inner and outer scales. The inner andnternal scales are identical and there is a simple multi-licative relation between the external and outer scales.he two lengths will be explicitly defined after furtherevelopment.Von Kármán [18] and von Kármán and Lin [19,20] sug-

ested a form for the velocity spectrum for ��0−1, which

as later revised to the temperature spectrum [16]Panchev’s sections 4.5, 6.2, and 6.7),

�T�� = 0.0330CT2 ��2 + �1.071/�0�2�−11/6, �2�

here 0.0330=5/ �18��1/3�� and 1.071= ��2/3�� /��5/6��3/2. The introduction of the constant

.0330CT2 seems to be the idea of Tatarskii [17] to reduce

he coefficient to a single strength of turbulence param-ter CT

2 . The temperature structure constant CT is definedrom the inertial subrange form of the structure functioni.e., for �0�r��0)

DT�r� = CT2r2/3. �3�

he introduction of the 1.071 coupled with �0 is attribut-ble to Reinhardt and Collins [8] to make the definition of0 more meaningful. The outer scale L0 is related to �0 by0=�0 /1.071 in order to make the outer scale definitiongree with the generally accepted one given by Strohbehn21].

To account for the internal scale effect at the dissipa-ive range, Golitsin [22] (reviewed by Panchev [16] in960) suggested a Gaussian roll-off

�T�� −11/3 exp�− ��/�M�2�, �4�

here �M=5.92/�0 in the present context is the internalcale wavenumber. Strohbehn [21] appears to be the firsto make the obvious connection between Eqs. (2) and (4)o get

�T�� = 0.0330CT2 ��2 + L0

−2�−11/6 exp�− ��/�M�2�. �5�

quation (5) is what is referred to in the laser propaga-ion literature as the modified von Kármán spectrum.ince for the most part this paper is only concerned with

nput and inertial range effects, we will concentrate onhe validity of the von Kármán spectrum given by Eq. (2).

We can now define internal and external scales �0 and0. The internal scale is the separation r at the intersec-

ion of DT�r��0� and DT��0�r��0�. This is consistentith Tatarskii [17], Section 1.13. At this point Tatarskii

17] also clarifies the definition of �0 with respect to theolmogorov microscales. The external scale is never

learly defined by Tatarskii [17], but in essence Panchev16] uses the precise definition that �0 is the separation rt the intersection of DT�r��0� and DT��0�r�.Taylor’s hypothesis [23], or frozen-flow assumption,

uggests the turbulence is simply being transported by axed point and is not changing over spatial scales of in-erest. It says that the spatial and temporal spectra, afterhe proper scaling with the transport velocity , are iden-ical. Panofsky et al. [24] state the hypothesis is “validlose to the ground” for lag distances up to 90 m provided

/�̃1/3, where is the wind speed standard deviation.his translates to equivalency of spectra for temporal fre-uencies f�̃ /570. We do not know what is meant byvalid” or “close to the ground.” Furthermore, the / cri-erion does not seem adequately restrictive. We do expecthe hypothesis to hold in the inertial subrange for fairlyomogeneous terrain.

. Theoretical Relations Based on von Kármán’spectrumhe first derivations are for the statistical description of

he temperature fluctuations and are based on the von

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ármán spectrum [8]. No internal scale effects are in-luded. The calculations and representative plots are forhe spatial structure function and these temporal spectra:emperature at a single probe and temperature differencet a probe pair. Based on these one could also compute theross spectrum and coherency.

The temperature structure function is defined as

DT�r� = ��T�r1� − T�r2��2�, �6�

here the brackets indicate an ensemble average and rr1−r2. Homogeneity and isotropy are assumed from

he start. From Tatarskii [17], Section 1.13, the structureunction is written in terms of the three-dimensionalpectrum

DT�r� = 8�0

�1 − sinc��r��T��2d�. �7�

fter the integration of Eq. (2) into Eq. (7),

DT�r� = CT2r2/3�1.0468�r/L0�−2/3

− 0.62029�r/L0�−1/3K−1/3�r/L0��, �8�

here K is the modified Bessel function of the third kind,th order [25,26], 1.0468=2�� / �3��5/6��, and 0.6202925/3�� / �3��5/6��1/3��. The two important asymptotesre

DT�r � �0� = CT2r2/3, �9�

DT��0 � r� = CT2�0

2/3. �10�

bviously the intersection of DT�r��0� and DT��0�r� ist r=�0. It was the advance knowledge of this intersec-ion that caused the insertion of 1.071 in the spectrumEq. (2)]. This normalized structure function is identicalo 1−C�r� where C�r� is the autocorrelation coefficient ofemperature.

Since authors use different definitions of spectra—ome one-sided, some bilateral, some with an extra 2�actor—we start by writing the spectrum to be used heren terms of the temporal covariance R�� in order to define�f�:

R�� =0

F�f�cos�2�f��df. �11�

atarskii [17] in Section 1.20 uses a two-sided spectrum,��, which is related to F by

F�f� = 4�W�2�f�. �12�

n that same section Tatarskii [17] relates the temporalnd spatial spectra

FT�f� = 8�2−12�f/

�T��d�. �13�

arrying out the straightforward mathematics we find

FT�f� = 0.07308CT22/3�f2 + f1

2�−5/6, �14�

here f1= / �2�L0� is a break frequency and 0.073072/ ��2��2/33��1/3��.

The temperature-difference spectrum F�T is now calcu-ated in a manner somewhat paralleling the phase-ifference calculations of Reinhardt and Collins [8] and oflifford [27]. Inverting a form equivalent to Eq. (11), oneas

F�T�f;r� = 2−

d� cos�2�f��R��,r�, �15�

here r is the vector separation of the two probes. Aftersing some geometry and the frozen-flow hypothesis, webtain

R��;r� = − DT�� +1

2DT�r + �� +

1

2DT�r − ��. �16�

f we assume the temperature field is homogeneous andsotropic and that the angle between r and � is �, then

R�T��;r� = − DT�� +1

2DT��r + � cos ��2 + �� sin ��2�

+1

2DT��r − � cos ��2 + �� sin ��2�, �17�

here r= r. Inserting Eq. (7) into Eq. (17) and, in turn,nto Eq. (15) and interchanging the order of integrationields

F�T�f;r� = 8�0

d��T��2−

d� cos�2�f��

��2 sinc��

− sinc��r + � cos ��2 + �� sin ��2�

− sinc��r − � cos ��2 + �� sin ��2��. �18�

aving done the � integration, we obtain

F�T�f;r� = 16�2−12�f/

d��T��1 − cos�2�fr−1 cos ��

�J0�r sin ��2 − �2�f/�2��. �19�

fter substitution of the von Kármán spectrum, Eq. (2),nd after replacing u2+1= �� / �2�f��2 we obtain

F�T�f;r� =40�

9��1/3�CT

2−1�2�f/�−5/3,

0

duu�u2 + 1 + �f1/f�2�−11/6

��1 − cos� f cos �

f2 J0� fu sin �

f2 � , �20�

here f1= / �2�L0� and f2= / �2�r� are the characteristicrequencies. After the final integration of Eq. (20) webtain

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F�T�f� = 3.1272CT2r5/3−1��=�/2

−5/3 �1 − cos� f cos �

f2

��1 − G�� , �21�

here

1 − G�� =5

3��

2 5/6 1

��11/6�K5/6��, �22�

�2 = �1 + �f1/f�2��f sin �/f2�2, �23�

�0�=0, and 3.1272=24� / �9��1/3��. Resonances occur inhe high frequencies f� f2 for � near 0 and � since, at leastheoretically, similar turbulence is seen at both sensorshen the wind is blowing along the sensors. Since the

ime lag of such a flow is r−1 cos �, the minimums in thepectrum occur at fN=N� /r�sec �, where N is an integer.n the real atmosphere, however, the wind speed and di-ection are not the fixed values required by fN. The smallariations that normally occur in and � are sufficient toash out the theoretical resonances. For example when=0 and f� f2, 1−cos� ��1. It is interesting to considerhe two regions that have simple power-law dependenciesor �=� /2:

F�T�f � f1,f2� = 5.829CT2r5/3/, �24�

F�T�f1,f2 � f� = 0.146162/3CT2 f−5/3. �25�

ence the difference spectrum approaches the constantiven by Eq. (24) as f→0 and has the high frequency de-endence [Eq. (25)] that is exactly twice the inertial rangeependence of FT given by Eq. (14) for f� f1:

f�T�f1,f2 � f� = 2FT�f1 � f�. �26�

n a similar fashion there are three regions when �=0 or:

F�T�f � f1,f2� = 3.1272CT2L0

5/3−1�f/f2�2/2, �27�

F�T�f1 � f � f2� = 3.1272CT2r5/3−1�f/f2�1/3/2, �28�

F�T�f1 � f� = 3.1272CT2r5/3−12�f/f2�−5/3 sin2�f/�2f2��.

�29�

ence the power law dependencies are f2, f1/3, and f−5/3 inhe three ranges. If the wind velocity is sufficiently fluc-uating that sin2� � is replaced by its average, 1/2, thenhe result is the same as Eq. (25).

. DATA COLLECTIONxperiments were run under nearly ideal conditions at

wo sites: the Verona, New York test site of the RADC;riffiss Air Force Base (AFB); and the laser test range ofir Force Weapons Laboratory (AFWL), Kirtland AFB,ew Mexico. By ideal we mean moderately high winds atidday, clear skies, and high turbulence strengths.To obtain spatial spectral information on microtem-

erature, nine microthermal probes were oriented in a

eometrical progression on a horizontal boom positionableto 8 m above the ground. Probes were spaced in a loga-

ithmic spacing so as to maximize the number of separa-ions relative to the number of probes. (Nominal probe-to-robe spacings along the boom were at 4, 5, 10, 20, 40, 80,60, and 320 cm. Using this scheme one can generate asany as 36 unique separations, taking one at a time, two

t a time, etc. Thus, for this example, the maximum sepa-ation is the sum of all of these: 639 cm.) Common meteo-ological observables recorded include wind speed, windirection, barometric pressure, and gross temperaturerom a dew-point system.

The microthermal sensors and amplifiers were basedn a design of Ochs at National Oceanic and Atmosphericdministration (NOAA) [28]. The sensors themselves areshort piece of Wollaston-process silver-on-platinum wirecid etched to give an active element 2 mm long and2.5 �m in diameter. These act as cold-wire probes so

ong as the wind velocity exceeds �0.5 m/s. Measure-ents of the response time of such a wire were reported

y Ochs [29] as 900 �s in still air and 300 �s in winds of5 m/s. Since the response is exponential, the sensor acts

s its own low-pass filter. The length of the sensor deter-ines the minimum spatial scale to which the probe can

espond. The high frequency roll-off will be seen later inresented spectra, and should not be taken as a measure-ent of inner scale. At the low end of the spectrum there

s a single high-pass filter set to 1.6�10−3 Hz to removehe effects of diurnal temperature changes that are notow frequency turbulence effects.

Since we were measuring spectral densities into theow frequencies, below 1 Hz, we were concerned with elec-ronic component drift in the bridge–amplifier circuit andith potential thermal drift in the probe itself. To empiri-

ally check the noise level of the electronic system, welaced a microthermal probe in an oil bath and comparedt with a nearby sensor in the air. We found the oil-mmersed probe to have a noise level at least 20 dB belowhe signal spectrum, merging with the air probe at theery highest frequencies when noise begins to dominate.hus electronics drift does not contribute to the low fre-uency spectra observed. A similar question about probehermal drift was addressed. Our probes were based onhe design and performance measurements of Ochs [28],ho states that as the probes draw less than 450 �A cur-

ent they are effectively cold-wire probes so long as theind velocity exceeds 0.5 m/s. Since solar loading is effec-

ively much smaller than any self-heating, these probesre cold-wire units for present purposes.

. MICROTHERMAL DATA COMPAREDITH VON KÁRMÁN’S MODEL

icrothermal data were collected at the Verona test site,riffiss AFB; essentially flat terrain in the center of theohawk Valley. Mission 210873 (August 21, 1973) was

un at midday during the most unstable conditions of tur-ulence with a clear sky and moderate winds. Conditionsnd gross statistics are tabulated in Table 1.Theoretical curves and the measured spectral and spa-

ial data are presented in Figs. 1–3. In Fig. 1 the averagedormalized structure function is plotted versus separa-

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ion r for 0.04�r�6 m. For separations r�̃0.6 m the dataoticeably deviate from the von Kármán curve. Thiseans there is a higher correlation between the tempera-

ures on probes in the vicinity of the external scale than isredicted. When translated into optical propagationerms this means a wider coherence function than previ-usly predicted. The data further indicate that a modifiedurve is needed that would depart from the inertial sub-ange at a smaller separation and have a more gradualnee in approaching the unity asymptote. At the least theata demonstrate the turbulence was Kolmogorov, withn r2/3 dependence in the inertial range.The spectra in Figs. 2 and 3 are for two sensors selected

uch that their separation, r=0.289 m, is in the inertialange. Figure 2 is a plot of the power spectra of the indi-idual temperature signals. Apparently the two sensorstatistically saw essentially the same fluctuations. Theheoretical curve there shows a discrepancy in the low fre-uencies for f�̃f1. This could have been predicted by Fig. 1ince large separations correspond to low frequencies.greement in the inertial subrange is excellent and ad-

Table 1. Conditions Recorde

onditions

loud cover 20%isibility 30 milesarometric pressure 1003 mbaremperature 22°Cround Dryensor height 2.26 mrass height 0.07 m

aWind angle 0 corresponds to wind blowing along the sensor array.

ig. 1. Normalized structure function data from mission 210873nd best theoretical fit using von Kármán’s spectrum. Note theivergence of data from theory at scales greater than 1/5 theuter scale.

quately shows the Kolmogorov power law, f5/3. In the fre-uencies 10� f�100 Hz there is slightly more power thann inertial subrange allows for. This can only be in theurbulence itself and can be attributed to this region ofhe spectrum being influenced by the approaching inter-al scale. At the high frequency end of the spectrum is thenticipated roll-off due to filtering and the physical size ofhe probe, possibly confounded by the onset of the dissa-ative regime (as discussed above); however there is andditional feature that deserves mention as it appearsere and in subsequent temporal spectral: the “bump”hat occurs at �30 Hz. This is not an artifact, as its pres-nce was argued by Hill in 1978 [30] latter having beeneen by Champagne et al. in 1977 [31]. The bump appearsn all our spectra.

The temperature difference spectrum is in Fig. 3 andhere the excess low frequency power can no longer be at-ributable to an error in the von Kármán spatial spec-rum. In fact a difference measure for �� /2 should be a

Griffiss AFB Mission 210873

Statistics

Wind velocity, (m/s) 2.25Wind standard deviation (m/s) 0.617

Wind anglea 55.6°Wind angle � (rad) 0.34Outer scale �0 (m) 4.22

CT2 �°C2 m−2/3� 0.086

/ 0.27

ig. 2. Normalized temperature spectral data and theoreticalurve based on von Kármán’s spectrum. (Abscissa is in units ofogarithm of frequency in hertz. This is the case for all temporalpectra shown here.) Individual spectra were normalized prior toveraging for this final data plot. Divergence of data from theoryn low frequencies is attributable to the spectral model, therozen-flow assumption, or both.

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airly smooth flat curve below f= f2. The sensor pair is notensitive to changes in the distribution of turbulent eddyizes for scales significantly longer than the separation.ather, the excess power seen in the data is attributable

o an inappropriateness of the frozen-flow hypothesis. Thealue of / for this mission is 0.27, so it is not surpris-ng that the hypothesis is not working well for scale sizesonger than 18 m in the difference spectrum.

In summary, there is obviously some inadequacy in theon Kármán spatial spectrum for wavenumbers in the in-ut range. The errors were translated into discrepanciesetween theory and data in the structure function for�̃�0 and in the spectra for f�̃f1. Section 5 will develop anmpirical model for the spatial spectrum to better matchhe data. Since one set of data is likely insufficient to es-ablish such a model we will then show how the revisedheory compares with the data of three other missions atwo radically different sites and at three altitudes.

. EMPIRICAL MODELINGt this point in the analysis we do not know how much of

he excess low frequency power is attributable to the usef von Kármán’s spectrum and how much to the invalidityf Taylor’s [23] hypothesis. Since the structure function isurely a spatial measurement not dependent on frozenow and since it is related directly to the spatial spectrumhrough Eq. (7), it will be the basis for the development ofnew model for the spatial spectrum. The model will be

hen treated as was the von Kármán form in that the tem-oral spectra will be derived and compared with the data.

ig. 3. Normalized temperature difference spectral data andheoretical curve based on von Kármán’s spectrum. (Again,bscissa is in units of logarithm of frequency in hertz.) Note thativergence in low frequencies is likely attributable to the frozen-ow assumption.

. General Form of the Modele assume the form of the spatial spectrum is

�T�� = c��L0�q + ��L0�s�−p, �30�

here once again the internal scale effects are ignored.he coefficient c will soon be related to CT

2 and to L0. Thisquation reduces to the von Kármán form when q=2, s0, p=11/6, and c=0.0330CT

2L011/3. The condition for finite

ower is that DT� �, equaling twice the temperatureariance, be finite:

DT� � = 8�0

�T��2d� � . �31�

his is satisfied if sp�3�qp, which in turn implies theower-law dependence −sp in the low wavenumbers is al-ays greater than the high frequency dependence −qp.his insures a knee in the curve in the vicinity of�L0

−1. After substituting Eq. (30) into Eq. (31) we get

DT� � =8�c

L03�q − s��p�

��3 − sp

q − s ��qp − 3

q − s . �32�

We will not attempt to adjust c so that DT� � is not aunction of s, p, and q. Rather c is set by the inertial sub-ange form of DT�r�. Since DT�r�L0� does not depend onhe low wavenumber shape of �T and since the totalower is finite, we can calculate DT�r�L0� by temporarilyssuming a form much simpler than Eq. (30),

�T�� = c��L0 + 1�−qp. �33�

fter substituting Eq. (33) into Eq. (7) we obtain a conflu-nt hypergeometric function, which is then expandedbout r=0. We keep only the lowest-order term:

DT�r � L0� = crqp−3L0−qp8�2

csc��2 − qp��

��qp − 1�sin��qp − 2�/2�.

�34�

ven though this is a direct result of using Eq. (33), thisquation applies for all qp�3. When qp is an integer,’Hospital’s rule must be used to evaluate Eq. (34). Ineeping with the accepted definition of CT

2 in Eq. (9), weet CT

*2 be the constant of proportionality in the inertialubrange

DT�r � L0� = CT*2rpq−3. �35�

comparison of Eq. (34) with Eq. (35) reveals that

c = �CT*2L0

pq/�8�2��pq − 1�sin��2 − pq��csc��pq − 2�/2�.

�36�

Another important quantity derived from the generalpectrum is the external scale length �0. In the case of aodified von Kármán spectrum �0=1.071L0. Once again,

he external scale is the value of r whereT�r�L0� /DT� �=1. We evaluate �0 from Eqs. (32) and

34):

tEf

a

Ttlth

BFdqwtet

Trmvw

F�qsi

Fmsls

F(btwlT


��0/L0�pq−3 = ��3 − sp

q − s ��qp − 3

q − s ��pq − 1�

�q − s��p�

�sin��2 − pq��csc��pq − 2�/2�. �37�

As for the general temporal spectrum, we can derivehe low and high frequency asymptotes. By substitutingq. (30) into Eq. (13) we find for qp�2, 0�sp�2, and� f1

L02

8�2cFT�f� →

1

�q − s��p��2 − sp

q − s ��qp − 2

q − s −

1

2 − sp� f

f1 2−sp

, �38�

nd for qp�2 and f� f1

L02

8�2cFT�f� →

1

�pq − 2�� f

f1 2−pq

. �39�

he requirement that sp�2 applies only to Eq. (38). Cer-ainly, the power is still finite for 2�sp�3, though nowimf→0FT�f�→ . A general form for the difference spec-rum F�T�f� is quite difficult to obtain analytically,owever, we know that F�T�f1 , f2� f�=2FT�f1� f�.

. Introduction of a Kolmogorov Inertial Rangeor the remainder of this paper the power-law depen-ence in the inertial subrange will be Kolmogorov:p=11/3. Hence c=0.0330CT

2L011/3. We are more concerned

ith the power spectra at low frequencies and the struc-ure function at large separations than with small differ-nces in the inertial range. The normalized spatial spec-rum �1/c��T�� versus �L0 is plotted in Fig. 4 for q=2.

ig. 4. Modeled temperature spatial spectrum for q=2, 0�sp2. Similar curves can be derived for other values of q: smaller

’s give a smoother transition, larger q’s a more abrupt knee (nothown). For any value of q (here q=2), an increase in sp resultsn a rounder transition at the knee of the curve.

he value q adjusts the rate of transition from the inputange to the inertial range. Higher q’s (not plotted) give aore abrupt transition and lower q’s a smoother one. The

alue of s used changes the power law, −sp, in the lowavenumbers.

ig. 5. Normalized temperature structure function for theodel when q is set to 2. Not shown, but as with the spectra,

maller q’s give a rounder knee; larger q’s a sharper knee. Simi-arly, small sp values give sharper knees; larger ones givemoother transitions.

ig. 6. Normalized temperature structure function dataGriffiss AFB mission 210873) and best-fit theoretical curveased on a proposed model with q=2, p=11/6, and s=1. Clearlyhere is a better fit than seen in Fig. 1, at least with this data set,ith a more gradual transition from the inertial subrange to

arge scales. Conditions for these data collection are given inable 1.

aisetdsts

trsvmmonTtsstviK

atltstm

6DSg

Fttv

F2pw

Fssfifla


Numerical integration is required to generate DT�r� forny q and sp—the two governing parameters. Since DT� �s given by Eq. (32), we need only to plot DT�r� /DT� � ver-us r /L0. This is shown in Fig. 5 with the same param-ters as in Fig. 4 (again q=2). The best fit of these curveso the data of mission 210873 is found by overlaying thatata and by sliding the experimental curve along the ab-cissa. Parameters q=2 and sp=11/6 seem to give the op-imum match. In Fig. 6 the structure function data of mis-ion 210873 is overlayed on that optimum curve.

The next step in validating the model is in the calcula-ion of the single-T power spectrum. Although we have al-eady chosen specific parameters to represent the finalpectrum, we still investigated the shapes of FT�f� forarious q, s, and p. Figure 7 shows the corresponding nor-alized theoretical spectra �L0

2 / �8�2c��FT�f� versus nor-alized frequency f / f1 (q=2 only, for brevity). The effect

f power in the spatial spectrum below �=2�f / is ig-ored because of the lower limit of the integral [Eq. (13)].hus there is not as pronounced a difference between theemporal spectra as there was with the spatial. There isufficient low frequency power, however, to follow theingle-T spectral data of mission 210873. Figure 8 showshat comparison and it is apparent how the parametricalues q=2 and s=1 have caused a more gradual breaknto the low frequency asymptote relative to the vonármán model.To complete the comparison between the new model

nd the data of mission 210873, we present theemperature-difference spectra in Fig. 9. There is veryittle noticeable difference between these curves andhose for the von Kármán spectrum. This is as expectedince spatial spectrum changes below �=L0

−1 are not no-iced in a probe separation r�L0. The remaining errorust be due to the use of Taylor’s hypothesis.

ig. 7. Normalized temperature temporal spectra (single T) forhe model with q=2. Earlier observations regarding the knee ofhe curve and the parameter q apply here as well, with smalleralues of sp giving a more abrupt knee.

. COMPARISON OF MODELS WITH OTHERATAince one mission at one site is not sufficient to develop aeneralized model, other data were collected at different

ig. 8. Single probe temperature spectral data (mission10873), and the theoretical model curve based on using q=2,=11/6, and s=1 (as with Fig. 6 where the structure functionas plotted). This demonstrates excellent fit.

ig. 9. Normalized temperature difference spectral data (mis-ion 210873), and the theoretical model curve for q=2, p=11/6,=1, and �=0. Agreement may seem to break down in the lowestrequencies, but since the difference spectrum is largely model-ndependent in these frequencies, we attribute this to the frozen-ow assumption breaking down in conditions of high wind speednd direction variations.

spTwat

ATowsoltlatat

tsK

clsf[

BUcfGdtFt

CTsfpspdsr

dttdhvsatb

7TTps

C

CVBTGSV

tor.

F2bs


ites at different heights above ground. These are com-ared with the new theory to show an equally good fit.he missions are: 200373 at Kirtland AFB where sensorsere at 20 and 33 m above uniform terrain, and 040973nd 011073 at Griffiss AFB under ideal conditions similaro 210873.

. Mission 200373o see if local terrain governed the low frequency behaviorf the spectra at Griffiss AFB, data taken at Kirtland AFBere processed for temporal spectral content only. Two

ensor pairs were positioned at 20 and 33 m above groundn a stable tower. The tower is near the center of a shal-ow canyon 1.6 km wide and with walls 43 and 27 m abovehe base of the tower. There is sparse desert vegetation—ow brush and juniper trees. The canyon floor is hard dirtnd is rocky. Winds are typically up or down canyon andhe ratio / is typically low. Whereas the nine sensorst Griffiss AFB are on a long manually rotatable boom,he sensor pairs at Kirtland AFB are on windvanes that

Table 2. Conditions Recorded

onditions 200373

loud cover Scatteredisibility 30 milesarometric pressure 811 mbaremperature 15.6°Cround Dryensor height 20, 33 megetation Minimal

aSensors on wind vane always aligned with probe line perpendicular to wind vec

ig. 10. Single probe temperature spectral data (mission00373, Kirtland AFB), 33 m altitude, and a theoretical curveased on the proposed model �q=2,p=11/6,s=1�. This demon-trates excellent data agreement with the model.

rack direction variations at �0.5 Hz with a thresholdpeed of 0.5 m/s. These conditions are summarized forirtland AFB in Table 2.Single-T spectra shown in Fig. 10 for the 33 m height

orroborate the use of the new model (22 m data are simi-ar). The temperature-difference spectrum (not plotted)till has the upturn in the data below the external scalerequency f1; again suggesting problems with the Taylor22] hypothesis for low frequency temperature spectra.

. Mission 040973nfortunately the Kirtland AFB mission could not in-

lude a measurement of the structure function, so to geturther backup spatially, another mission was run atriffiss AFB for structure function data only, for the con-itions noted in Table 3. The normalized structure func-ion, measured with seven sensors operative, is plotted inig. 11. The fit of the data to the modeled curve is cer-

ainly consistent with the fit of mission 210873.

. Mission 011073his data collection mission at Griffiss AFB had low windpeed and direction variances and thus we expect therozen-flow hypothesis to work better. (See conditions re-orted in Table 3.) The measured structure function andpectral data in Figs. 12–14 show the same behavior asreviously presented missions. One new aspect is that theifference spectrum below the external-scale frequencyhows an excellent fit to the theoretical curve. This is aesult of low values of / and �.

An additional observation is that for all spatial spectralata, the best-fit outer scale is approximately 1.2 to 2imes the height above ground, and usually closer to 2imes (see Tables 1–3). As discussed in the Introduction,ominant large scale sizes for turbulence spectra areeight above ground and height of the boundary layer in-ersion. Since all data reduced here are for daytime un-table conditions at heights of 2–33 m, clearly the heightbove ground is the dominant scale length, and havinghe outer scale proportional to sensor height could haveeen anticipated.

. SUMMARY OF EQUATIONS FORHE NEW MODELhis section summarizes the equations for the newarameters q=2, s=1, and p=11/6. The new spatialpectrum is

irtland AFB Mission 200373

Statistics 200373

Wind velocity, (m/s) 6.52Wind standard deviation (m/s) 1.38

Wind anglea 90°Wind angle � (rad) 0.25Outer scale �0 (m) 37.7

CT2 �°C2 m−2/3� 0.0409

/ 0.212

for K

TE

A

Tt=v

fnm

wt

Tt[

oEcsvcaf

C

CVBTGSG

Ftap

FmA


�T�� = 0.0030CT2 ��2 + �/L0�−11/6. �40�

he structure function is calculated by the substitution ofq. (40) into Eq. (7):

DT�r� = CT2r2/3�1.1078�r/L0�−2/3 − 4.6173�r/L0�−5/3

�Im U�1/6,− 2/3,− ir/L0��. �41�

dditionally,

DT�r � L0� = CT2r2/3, DT�L0 � r� = 1.1078L0

2/3CT2 ,

DT�r � L0�/DT�L0 � r� = 0.9027�r/L0�2/3.

he function U�a ,b ,c� is a confluent hypergeometric func-ion [25]. The external scale �0 in terms of L0 is �01.1660L0, not much different from �0=1.071L0 for theon Kármán spectrum.

The single-T power spectrum does not have as neat aorm as the structure function and is best obtained by theumerical integration of Eq. (40) into Eq. (13). Formally itay be represented as

Table 3. Conditions Recorded for G

onditions 040973 010173

loud cover 0% 20%isibility 30 miles 30 milesarometric pressure 1005 mbar 1028 mbaremperature 32°C 22°Cround Dry Dryensor height 2.26 m 2.26 mrass height 0.07 m 0.07 m

aWind angle 0 corresponds to wind blowing along the sensor array.

ig. 11. Normalized temperature structure function data andheoretical model curve (mission 040973, Griffiss AFB). Excellentgreement is found with the parameters chosen earlier: q=2,=11/6, s=1.

FT�f� = 0.07307CT22/3f−5/3

2F1�11/6,5/3;8/3;− f1/f�,

�42�

here 2F1�a ,b ;c ;z� is a hypergeometric function [25]. Thewo asymptotes for FT�f� are

limf→0

FT�f� = 13.919CT2−1L0

5/3�1 − 1.1232�f1/f�−1/6�, �43�

limf→

FT�f� = 0.07307CT22/3f−5/3. �44�

he inertial subrange dependence [Eq. (44)] is preciselyhe same as the dependence for the von Kármán spectrumEq. (14)], as it must be.

The temperature-difference spectrum can only bebtained by the numerical integration of Eq. (40) intoq. (19). The two asymptotes f� �f1 , f2� and f� �f1 , f2� are

alculable. When f� f1 the modified portion of the spatialpectrum has been ignored and the form derived from theon Kármán spectrum is applicable. This is Eq. (21) ex-ept that now in Eq. (23) � is given by �f sin �� / f2. Thesymptote for small f where also ��0 and r sin ��L0 isound via Erdelyi [26], transform pair 10, p. 23:

s AFB Missions 040973 and 011073

Statistics 040973 011073

Wind velocity, (m/s) 1.62 3.16ind standard deviation (m/s) 0.49 0.78

Wind anglea 116° 95°Wind angle � (rad) 0.43 0.33Outer scale �0 (m) 2.92 5.11

CT2 �°C2 m−2/3� 0.142 0.26

/ 0.30 0.25

ig. 12. Normalized temperature structure function data andodel curve for q=2, p=11/6, and s=1 (mission 011073, GriffissFB), showing the best agreement of all the data shown.

riffis

W

Tn

fs

8Wmsssspstffned

aEsstwtdsat

ralamdcgKtpttdttacbdts

mepttstm

Fq

Fqlav


limf→0

F�T�f� = 5.8276−1CT2 �r sin ��5/3

��1 − 1.045�rL0−1 sin ��1/3�. �45�

his assures us that the modifications to the theory doot account for an excess low frequency power in the dif-

ig. 13. Single-probe temperature spectral data and model for=2, p=11/6, and s=1 (mission 011073, Griffiss AFB).

ig. 14. Temperature-difference spectral data and model for=2, p=11/6, and s=1. Note excellent agreement even in the

owest frequencies, well below f1, which is attributable to the bestpplicability of the model in low wind speed and directionariations.

erence spectrum, since this is identical to Eq. (21) for theame conditions.

. CONCLUSIONSe have developed an empirical model for the spatialicrotemperature spectrum, validated for unstable atmo-

pheric conditions. Rather than rely solely on temporalpectra (and thus the frozen-flow hypothesis), the spatialtructure function, related via a Fourier transform to thepectrum, was measured directly. A spectral model wasroposed and the parameters varied until a best fit to thetructure function data was obtained. The best model washen transformed to the temporal spectrum using therozen-flow hypothesis and again good agreement wasound. The only residual error traceable to inappropriate-ess of frozen flow was in the difference spectrum andven that error was reduced when the variances of windirection and speed lessened.Four data-gathering missions were run at the Kirtland

nd Griffiss AFB sites, both over rather uniform terrain.ven for substantially different conditions at these twoites, the data fit the new model well. An inspection of thetructure function data at Griffiss AFB revealed no no-iceable dependence of curve shape or external scale onind direction with respect to the sensor line. Presently,

here is no proper experiment to measure the three-imensional external scale since sensor wake effects areignificant when a measurement along the wind vector isttempted and since the external scale varies with alti-ude in a vertical measurement.

All data exhibited Kolmogorov behavior in the inertialanges due to running under the best conditions for thevailable sites: moderate winds, well-developed turbu-ence and convection at midday, good mixing, few clouds,nd low wind direction fluctuations. A different behavioray be encountered at the near-neutral conditions of

awn and dusk and in the stable conditions usually en-ountered at night, but these conditions were not investi-ated. Measurements made by Kaimal et al. [32] in aansas wheat field indicate the more nearly stable condi-

ions may fit the von Kármán model rather than the pro-osed one. However, in unstable conditions the data seemo support the new model. The seeming discrepancy be-ween stable and unstable conditions can be attributed toiffering boundary layer conditions but this requires fur-her research. Further experimentation is needed to de-ermine if the low frequency behavior depends on Rich-rdson’s number or on the temperature structureonstant. Under some conditions the turbulence may evene non-Kolmogorov. It is hoped the generality in the earlyevelopment of the model will allow the reader to extendhis model to his particular case if necessary, including totable atmospheric conditions.

The proposed model seems to make the mathematicsore complicated, but it does not introduce new math-

matical difficulties such as nonconvergence. Thoseropagation calculations that depend on large scales ofurbulence, such as beam wander, should be redone usinghe new model. If the calculations do not depend on largecales but require a formula that has finite power, thenhe von Kármán spectrum should be used as it is easier toanipulate.

ttwTKt

ATdLsstept(tbf

FdS

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3


In conclusion, the model spectrum intends to addresshe transition scaling from the proven Kolmogorov regimeo that of large scale, including the transition regimehose scale is dominated by the height above ground.his is intended to be an improvement on the vonármán spectrum with its flat low frequency dependence

hat does not address the transition.

CKNOWLEDGMENTShe work presented is a result of in-house efforts con-ucted in 1973–1974 supporting DARPA’s High Energyaser Program. The authors thank David Youmans, Rus-ell McGregor, and John Bradham of RADC for their as-istance in data acquisition, processing, and interpreta-ion. Also, the cooperation of Robert Endlich of AFWL wasssential in running and processing the Kirtland AFBhase of the missions. John Wyngaard and J. C. Kaimal ofhe U.S. Air Force Cambridge Research LaboratoryAFCRL) Boundary Layer Branch assisted in ensuringhat the experiments were proper and in providing theasic computer software. Finally, thanks to two reviewersor exceptionally helpful observations and suggestions.

This work was sponsored by the United States Airorce. The views expressed are those of the authors ando not reflect the official policy or procedure of the Unitedtates Government.

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proposed form for the atmospheric turbulence spatial spectrum at large scales

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