Calculating the Microstructure ofAtmospheric Optical Turbulence

ARL-MR-419 December 1998

Arnold Tunick

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Calculating the Microstructureof Atmospheric OpticalTurbulence

ARL-MR-419 December 1998

Army Research LaboratoryAdelphi, MD 20783-1197

Arnold Tunick Information Science and Technology Directorate

Approved for public release; distribution unlimited.

Turbulent fluctuations in air density can cause significant distortionsof an electromagnetic signal or image. Density fluctuations can bedescribed in terms of air temperature, air pressure, water vapor, andCO2 content. We can calculate the refractive index structure constant,Cn

2, with the fine-scale dynamics of heat, moisture, and momentumdiffusion. This helps us to quantify the intensity ofturbulence-induced refraction. A better understanding ofturbulence-induced refraction can provide a means of evaluatingsensors under various atmospheric conditions or be used in thedevelopment of turbulence-compensation adaptive optic systems.This paper annotates one set of equations for the refractive indexstructure constant, C n

Abstract

2, taken from the literature.

ii

iii

Contents

1. Introduction ........................................................................................................................................... 1

2. Model Equations ................................................................................................................................... 32.1 Refractive Index Structure Constant .............................................................................................. 32.2 Refractive Index of Air and Its Partial Derivatives ........................................................................ 4

3. A Model of the Microstructure of Atmospheric Optical Turbulence ......................................... 7

4. Summary ................................................................................................................................................ 8

References .................................................................................................................................................. 9

Distribution ............................................................................................................................................. 11

Report Documentation Page ................................................................................................................. 17

Figure

1. CN2 model output compared to scintillometer data taken at 2 m over a horizontalpath of 450 m........................................................................................................................................ 2

Tables

1. Microstructure of atmospheric optical turbulence model physical constants .............................. 72. Microstructure of atmospheric optical turbulence model input and output ................................ 8

1

1. IntroductionAtmospheric optical turbulence can modify the refractive index in air in away that can significantly alter the transmission and propagation of anelectromagnetic image or signal [1]. Even through weak turbulence, alaser beam can become highly scintillated and exhibit strong intensityfluctuations if propagated over a long distance [2]. Also, optical turbu-lence can reduce the efficiency of laser systems propagated from theground to an object in space [3]. In this regard, Walters [4] presented theresults of an investigation to develop a data-reduction algorithm forsequences of balloon-borne data aimed at providing vertical profiles ofthe refractive index structure constant. Walters asserts that knowledge ofboth turbulence and wind speed profiles could be helpful for the develop-ment of turbulence-compensation adaptive optic systems.

Andreas [5] defines the problem of estimating the refractive index struc-ture constant from meteorological point measurements and raises thequestion of whether or not point measurements can be used to predict apath-averaged assessment of turbulence-induced refraction. He citesDavidson et al [6] for an example of a bulk-layer method applied toestimating overwater optical turbulence. Also, Tunick et al [7] reported onan experiment wherein radiation and energy budget-derived turbulencedata are compared to scintillometer-retrieved data taken over a bare soilpath of 450 m. The semi-empirical models presented in Rachele andTunick [8] and Tunick [9] have also made estimates of the refractive indexstructure constant for comparison to observed turbulence (scintillometer)data. However, these types of first-order difference routines can result insignificant and sometimes extreme errors when point data in space andtime are used to represent area or path averages (see fig. 1), particularlythroughout periods before or after sunrise and sunset.

However, with increasing interest in high-performance computing,modeling the intensity turbulence-induced refraction is beingre-investigated through the use of large eddy simulations [10,11]. Therefractive index structure constant is recast as a variable that can bedetermined locally, given values for the heat and momentum flux andgradients of pressure, temperature, wind speed, and specific humidity. Asa means of illustrating a calculation of atmospheric optical turbulence ofthis type, an algorithm could be derived from equations collated fromdifferent articles in the open literature and tested using field experimentdata. In this report, the algorithm MAOT (Microstructure of AtmosphericOptical Turbulence), derived from equations collated from differentarticles in the open literature, is presented and the calculation is testedusing data generated from observations [12,7].

2

–12 –8 –4 0 4 8 12 16 2010–23

10–22

10–21

10–20

10–19

10–18

10–17

10–16

10–15

10–14

10–13

10–12

10–11

Cn

(m–2

/3)

Local time

2

Scintillometer dataCN2 model (input data: 1 and 4 m)CN2 model (input data: 2 and 4 m)CN2 model (input data: 4 and 8 m)

Source: A. Tunick, The Refractive Index Structure Parameter/Atmospheric Optical Turbulence Model: CN2, U.S. Army Research Laboratory, ARL-TR-1615 (1998).

Figure 1. CN2 modeloutput compared toscintillometer datataken at 2 m over ahorizontal path of450 m.

3

2. Model Equations

2.1 Refractive Index Structure Constant

Hill [13] gives an expression for the refractive index structure constant as

Cn

2 =n(x) – n(x + r) 2

r2 32 3=

Dn(r)r2 32 3

, (1)

where n is the refractive index in air, (x) and (x + r) denote position inspace, and the ensemble mean variance n(x) – n(x + r) 2 is the scalarstructure function. Batchelor [14] gives a connection between the function,Dn(r), in r space, and the spectrum for the scalar, Γn(k), in k space as

Dn(r) = 2 1 – sin (rk)

rkΓn(k) dk ,

0

∞(2)

where k is the wave number. Through dimensional analysis, the scalarstructure function and the scalar spectrum can be expressed in terms ofthe dissipation rate of turbulent kinetic energy, ε, and the diffusive dissi-pation rate of the scalar variance, χn. Hill [13] gives χn as

χn = 2 dn ∇n 2 , (3)

where, dn ≈ dh, assuming that the diffusion coefficients for the scalarsrefractive index and potential temperature are effectively the same [15].Then

dn = dh = w′ θ′ / ∂θ/∂z . (4)

The variable ε can be expressed as

ε = g/θ w′ θ′ – u′ w′ ∂U/∂z , (5)

where g is the acceleration due to gravity, u′ w′ is the ensemble meaneddy transport of horizontal momentum, w′ θ′ is the ensemble meankinetic heat flux, ∂θ/∂z is the vertical gradient of potential temperature,and ∂U/∂z is the vertical gradient of the total horizontal wind [16].

The resulting expressions for the scalar structure function and the scalarspectrum are

Dn(r) = b n χn ε– 1 3– 1 3 r2 32 3 , (6)

and

Γn(k) = β n χn ε– 1 3– 1 3 k– 5 3– 5 3 , (7)

4

given the Kolmogorov 2/3- and –5/3-dependencies for r and k, respec-tively, where bn and βn are constants (Hill [17,18] gives βn = 0.72). Whenequations (6) and (7) are substituted into equation (2), bn is given as

bn = – 65 βn

cos (x)(x)

5 35 3dx = 9

10 Γ(1/3) β n . (8)

Relationships among the gamma functions, Γ(p) , for 0 < p < 1 are givenby Weast et al [19]. Finally, the expression for the structure constant inequation (1) can be rewritten as

Cn2 = 2 bn dh ε– 1 3– 1 3 ∂n/∂z 2 , (9)

where bn = 1.736.

2.2 Refractive Index of Air and Its Partial Derivatives

The refractive index of air for the visible and near-infrared (3650 to6328 Å) region of the electromagnetic spectrum is expressed [20,21] interms of wavelength (in micrometers), barometric pressure (P, in milli-bars), temperature (T, in degrees Kelvin), and vapor pressure (e, thepartial pressure of the atmosphere due to water vapor content, also inmillibars) in the form presented by Andreas [5]:

nvi = 1.0 + m1PT + m1 + m2

eT × 10– 6 , (10)

where temperature, T, is defined as T = θ (P/Ps)(2⁄7); Ps is normally de-fined as sea level barometric pressure,

m1 = 23.7134 + 6839.397130.0 – σ 2 + 45.473

38.9 – σ 2 , (11)

and

m2 = 64.8731 + 0.58058 σ2 – 0.007115 σ 4 + 0.0008851 σ 6 , (12)

where σ = 1.0/λ (µm–1).

In the infrared region of the electomagnetic spectrum from 78,000 to190,000 Å [20,22], the refractive index is expressed in the form

nir = 1.0 + m1P – e

T + nirw × 10– 6 , (13)

where the refractive index of water vapor is given as

nirw = Q

957.0 – 928.0 T/To0.4 (X – 1.0)

1.03 T/To0.17 – 19.8X2 + 8.2X4 – 1.7X8 + 3.747 × 106

12,449.0 –X2 , (14)

5

where Q = 0.2166847 e⁄T, absolute humidity is in kg/m3, and

X =10.0 (µm)

λ (µm). (15)

In equations (10) and (13), vapor pressure, e, in millibars, can beexpressed [23] in terms of specific humidity, q, in units of grams of watervapor content per kilogram of moist air (dry air and water vapor com-bined) in the following form:

e =P q

mw/ma + 1 – mw/ma q, (16)

where specific humidity is defined [24] as

q =es m w/m a

PRH

100.0exp

m wL v

R*1.0T o

– 1.0T

, (17)

where es = 6.1078 mbar is the saturation vapor pressure at 0.0 °C; mw andma are the molecular weights of water vapor and of dry air, respectively;Lv = 2.5008 × 106 + 2.3 × 103 T (T in degrees Celsius) is the latent heat ofvaporization; R* = 8314.32 J °K–1kmol–1 is the universal gas constant; andRH is relative humidity in percent.

The derivatives of the refractive index given by equations (10) and (13)take the form

∂n∂z = ∂n

∂T∂T∂z + ∂n

∂e∂e∂z , (18)

so that

∂nvi∂T = – m1

PT2 – m2 – m1

eT2

∂e∂T × 10– 6 , (19)

where

∂e∂T = – 1.0

T2L v

R*/mwexp

L vR*/mw

1.0To

– 1.0T × e2 – e

P q . (20)

The partial derivative of n with respect to vapor pressure takes the form

∂nvi∂e

=m 2 – m 1

T× 10– 6 . (21)

The partial derivative of vapor pressure takes the form

∂e∂z

= ∂e∂q

∂q∂z , (22)

6

where

∂e∂q

=m w/m a P

m w/m a + 1.0 – m w/m a q 2. (23)

The partial derivative of T in terms of the scalar potential temperaturetakes the form

∂T∂z

=

∂θ∂z

+ 2.07.0

TP s

P 2

P s

P

– 57

– 57 ∂P

∂z

P s

P

2 72 7.

(24)

Equations (13) and (14) can be rewritten as

nir = 1 + m1PT + 0.21668 f (T,X) – m1

eT × 10– 6 , (25)

where

[A] =

957.0 – 928.0 T/To0.4 (X – 1.0)

1.03 T/To0.17 – 19.8X2 + 8.2X4 – 1.7X8 + 3.747 × 106

12449.0 – X2 . (26)

The partial derivative of nir with respect to temperature can now take theform

∂nir∂T = – m1

PT2 – e

T2 0.21668[A] – m1∂[A]∂T × 10– 6 , (27)

where

∂[A]∂T

= –

0.1751T o

TT o

– 0.83957.0 – 928.0 T

T o

0.4(X – 1.0)

1.03 TT o

0.17– 19.8X 2 + 8.2X 4 – 1.7X 8

2

–

371.2273.15

TT o

0.6

1.03 TT o

0.17– 19.8X 2 + 8.2X 4 – 1.7X 8

.

(28)

Lastly, the partial derivative of nir with respect to vapor pressure takes theform

∂nir∂e =

0.21668[A] – m1T × 10– 6 . (29)

7

3. A Model of the Microstructure of Atmospheric OpticalTurbulence

The equations presented in sections 2.1 and 2.2 were programmed inFORTRAN to produce a computer model called MAOT. The MAOTmodel computes the refractive index structure constant, given values forthe heat and momentum flux and gradients of pressure, temperature,wind speed, and specific humidity as input. Table 1 gives values for themodel’s physical constants. Table 2 gives the results of testing the MAOTcalculation for different conditions of atmospheric stability. The modelinput is generated from observed surface layer data that were reported byTunick et al [7] except for the last column, which was derived from themicrometeorological data reported by Stenmark and Drury [12].

Values of Cn2 have been generally observed to range from about 10–12 to

10–16 m–2/3. The values of Cn2 for the column labeled Unstable (approxi-

mately 10–12 m–2/3) imply that the turbulence is intense, and considerableimage blurring or signal distortion could occur (similar to that seen whenone looks over an open field or a paved lot on a hot day). In contrast,the values of Cn

2 for the column labeled Weakly stable (approximately10–16 m–2/3 ) imply that the intensity of the optical turbulence might beconsidered negligible, except for where a light beam is transmitted over along distance. Higher values of Cn

2 given in table 1 correlate with higher(absolute) values of kinematic heat flux and potential temperature gradi-ent. The lower values of Cn

2 given in table 1 correlate with higher valuesof momentum flux and wind speed gradient. This observation makes thepoint that surface layer stability and turbulence are generally lessened bythe effects of wind shear and surface stress.

Parameter Symbol Unit Amount

Kolmogorov or Corrsin constant bn — 1.736Acceleration due to gravity g m/s2 9.8Temperature scaling To °K 273.15Molecular weight of water vapor mw g/mol 18.016Molecular weight of dry air ma g/mol 28.966Universal gas constant R* J °K–1 kmol–1 8314.32Saturation vapor pressure at 0.0 °C es mbar 6.1078Reference level pressure Ps mbar 1013.25

Table 1.Microstructure ofatmospheric opticalturbulence modelphysical constants.

8

Table 2. Microstructure of atmospheric optical turbulence model input and output.*

Condition of atmospheric stability

Parameter Unit Unstable Weakly unstable Weakly stable Stable

Kinematic heat flux °K m/s –0.470 –0.055 0.014 0.073Momentum flux m2/s2 –0.164 –0.217 –0.042 –0.191Potential temperature gradient °K/m –0.670 –0.124 0.010 0.310

Wind speed gradient ms /m 0.330 0.510 0.303 0.812

Specific humidity gradient gg /m –1.133 × 10–4 –6.667 × 10–6 1.000 × 10–4 –2.500 × 10–4

Pressure gradient mbar/m –0.10 –0.10 –0.10 –0.10

Model outputCn

2 visible m–2/3 1.633 × 10–12 3.382 × 10–14 8.590 × 10–16 1.176 × 10–13

Cn2 IR m–2/3 1.763 × 10–12 3.479 × 10–14 5.283 × 10–15 1.154 × 10–13

*Electromagnetic wavelength—visible 0.94 µmElectromagnetic wavelength—IR 10.6 µm

4. SummaryThe propagation of a light beam through the atmosphere is affected byrandom fluctuations in the refractive index of air [24] and it is thesefluctuations or discontinuities that cause optical turbulence. The refrac-tive index structure parameter is the quantitative measure for such turbu-lence. In this report, I have presented the algorithm MAOT, derived fromequations collated from different articles in the open literature. TheMAOT calculation was tested using kinematic heat flux and momentumflux data generated from observations. MAOT was regarded as a steptaken toward enhancing calculations of refractivity in the surface layerthrough the diurnal cycle.

9

References1. Burk, S. D., and W. T. Thompson, “Mesoscale modeling of summertime

refractive conditions in the southern California Light,” J. Appl. Meteor. 36(1997), pp 22–31.

2. Primmerman, C. A., T. R. Price, R. A. Humphreys, B. G. Zollars, H. T.Barclay, and J. Herrmann, “Atmospheric-compensation experiments instrong-scintillation conditions,” Appl. Opt. 34 (1995), pp 2081–2088.

3. Fouche, D. G., C. H. Higgs, and C. F. Pearson, “Scaled AtmosphericBlooming Experiments (SABLE),” Lincoln Laboratory Journal 5 (2) (1992),pp 273–293.

4. Walters, D. L., “Measurements of optical turbulence with higher-orderstructure functions,” Appl. Opt. 34 (1995), pp 1591–1597.

5. Andreas, E. L., “Estimating Cn2 over snow and sea ice from meteorological

data,” J. Opt. Soc. Am. 5 (1988), pp 481–495.

6. Davidson, K. L, G. E. Schacher, C. W. Fairall, and A. K. Goroch, “Verifica-tion of the bulk layer method for calculating overwater optical turbu-lence,” Appl. Opt. 20 (1981), pp 2919–2924.

7. Tunick, A., H. Rachele, F. V. Hansen, T. A. Howell, J. L. Steiner, A.D.Schneider, and S. R. Evett, “REBAL ’92—A Cooperative Radiation andEnergy Balance Field Study for Imagery and EM Propagation,” Bull. Am.Meteorol. Soc. 75 (1994), pp 421–430.

8. Rachele, H., and A. Tunick, “Energy balance model for imagery andelectromagnetic propagation,” J. Appl. Meteor. 33 (1994), pp 964–976.

9. Tunick, A., The Refractive Index Structure Parameter/Atmospheric OpticalTurbulence Model: CN2, U.S. Army Research Laboratory, ARL-TR-1615,(1998).

10. Peltier, L. J., and J. C. Wyngaard, “Structure-function parameters in theconvective boundary layer from large-eddy simulation,” J. Atmos. Sci. 52(1995), pp 3641–3660.

11. Wyngaard, J. C., N. Seaman, K. Gilbert, and M. Otte, The refractivitystructure of the lower atmosphere: connecting turbulence and meterology, to bepublished in Proceedings of the 1998 Battlespace Atmospheric and CloudImpacts on Military Operations (BACIMO) Conference, Hanscom AirForce Base, MA (1–3 December 1998).

12. Stenmark, E. B., and L. D. Drury, Micrometeorological field data from Davis,California: 1966–1967 runs under non-advective conditions, U.S. Army Tech-nical Report, ECOM-6051 (1970), p 604 (available from U.S. Army Re-search Laboratory, Adelphi, MD).

10

13. Hill, R. J., “Structure functions and spectra of scalar quantities in theinertial-convective and viscous-convective ranges of turbulence,” J.Atmos. Sci. 46 (1989), pp 2245–2251.

14. Batchelor, G. K., “Small-scale variation of convective quantities liketemperature in a turbulent fluid, Part 1: General discussion and the caseof small conductivity,” J. Fluid Mech. 5 (1959), pp 113–133.

15. Tatarski, V. I., The Effects of the Turbulent Atmosphere on Wave Propagation,Israel Program for Scientific Translations, Jerusalem (1971), p 472, (avail-able as NTIS Technical Translation 68-50464).

16. Wyngaard, J. C., “On surface layer turbulence,” Workshop on Micrometeo-rology, D.A. Haugen, ed., Amer. Meteor. Soc. (1973), p 392.

17. Hill, R. J., “Models of the scalar spectrum for turbulent advection,” J.Fluid Mech. 88 (1978), pp 541–562.

18. Hill, R. J., “Spectra of fluctuations in refractivity, temperature, humidity,and the temperature-humidity cospectrum in the inertial and dissipationranges,” Radio Sci. 13 (1978), pp 953–961.

19. Weast, R. C., S. M. Selby, and C. D. Hodgman, eds., Handbook of Chemistryand Physics, 45th Edition, The Chemical Rubber Co., Cleveland, OH (1964),p A-142.

20. Owens, J. C., “Optical refractive index of air: dependence on pressure,temperature and composition,” Appl. Opt. 6 (1967), pp 51–59.

21. Ciddor, P. E., “Refractive index of air: new equations for the visible andnear infrared,” Appl. Opt. 35 (1996), pp 1566–1573.

22. Hill, R. J., and R. S. Lawrence, “Refractive index of water vapor in infra-red windows,” Infrared Phys. 26 (1986), pp 371–376.

23. Gill, A., Atmosphere-Ocean Dynamics, Academic Press (1982), p 662.

24. Kunkel, K. E., D. L. Walters, and G. A. Ely, “Behavior of the TemperatureStructure Parameter in a Desert Basin,” J. Appl. Meteorol. 20 (1981), pp130–136.

11

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Form ApprovedOMB No. 0704-0188

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102

Calculating the Microstructure of Atmospheric OpticalTurbulence

December 1998 March 1998 to August 1998

Turbulent fluctuations in air density can cause significant distortions of an electromagnetic signal or image.Density fluctuations can be described in terms of air temperature, air pressure, water vapor, and CO2 content.We can calculate the refractive index structure constant, Cn

2, with the fine-scale dynamics of heat, moisture, andmomentum diffusion. This helps us to quantify the intensity of turbulence-induced refraction. A betterunderstanding of turbulence-induced refraction can provide a means of evaluating sensors under variousatmospheric conditions or be used in the development of turbulence-compensation adaptive optic systems. Thispaper annotates one set of equations for the refractive index structure constant, Cn

2, taken from the literature.

Refractive index, heat flux

Unclassified

ARL-MR-419

7FEJ706110253A11

B53A61102A

ARL PR:AMS code:

DA PR:PE:

Approved for public release; distributionunlimited.

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Unclassified Unclassified UL

2800 Powder Mill RoadAdelphi, MD 20783-1197

U.S. Army Research LaboratoryAttn: AMSRL-IS-EE atunick@arl.mil

U.S. Army Research Laboratory

17

Arnold Tunick

email:2800 Powder Mill RoadAdelphi, MD 20783-1197

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