Turbulence Simulation: Turbulence Simulation: 

Outer Scale Effects on the Outer Scale Effects on the 

Refractive Index Spectrum Refractive Index Spectrum 

By 

David H. Tofsted David H. Tofsted 

Computational and Informational Sciences Directorate Computational and Informational Sciences Directorate 

Battlefield Environment Division 

ARL-TR-548November 2000 November 2000 November 2000 

 

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Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum 

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D. Tofsted 

 

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13. ABSTRACT (Maximum 200 words) 13. ABSTRACT (Maximum 200 words) 

 

An analysis of the commonly used refractive index power spectrum indicates that its method of simulating outer scale 

effects is deficient in two aspects: it approaches a nonzero value at zero frequency and is ambiguous regarding definition 

and usage of the outer scale in its functional form. To resolve these problems, a modified form of the refractive index 

spectrum is proposed which properly models the behavior of the spectrum at zero frequency and simulates the energycontaining eddy region. In support of this analysis, outer scale is defined such that it is independent of the mathematical 

form used to describe the refractive index spectrum. Based on reported experimental data, a new equation for outer scale 

as a function of stability conditions is proposed, revealing dependence of the outer scale on both height above the surface 

and the stability-related Monin-Obukhov length. 

 

 

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propagation, optical turbulence, coherence diameter, outer scale, Kolmogorov spectrum, von 

Karman spectrum. 

 

 

 

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Preface

The structural eects of the outer scale of turbulence are signicant in certain

propagation problems in the Earth's lower atmosphere. In particular, image

wander (angle of arrival 
uctuations of distant ob jects) and shape distortion

of ob jects due to nonisoplanatism are critically dependent on the shape of

the refractive index power spectrum in its energy-containing-eddy region. To

correctly model these eects requires an accurate model of the outer scale of

turbulence, both as a function of height and as regards how it impacts the

refractive index spectrum.

In the past, the outer scale was dened rather vaguely, using order of magnitude

arguments. Or, at best, outer scale was dened based of the mathematical form

used in describing the refractive index spectrum itself. Also, the outer scale was

often equated with the so-called mixing length used in numerical weather model

closure methods for solving the Navier-Stokes equations.

But the mixing length is merely a convenient order-of-magnitude length

scale. Atmospheric optics has tended to adopt this value in lieu of better

parameterizations. This report attempts to ll this data void by dealing with

several aspects of the outer scale problem. To accurately determine outer

scale eects with regard to propagation problems involving optical turbulence

(turbulent image wander, image shape distortion, etc.) better models must be

sought than those based on reasonable guesses.

First, I propose a denition for the outer scale which is independent of the

particular mathematical form used in describing the refractive index spectrum.

Second, an analysis of temperature spectrum data from the 1968 Kansas

experiment is discussed that leads to a new form for the refractive index

spectrum. For completeness, I also propose a combined spectrum called the

composite spectrum that includes both inner and outer scale in
uences.

Third, based on the analysis of the spectrum itself, a relationship is determined

between the peak frequency data from the Kansas experiment and the outer scale

of turbulence. This relationship describes a functional form for the turbulence

outer scale as a function of both height above the surface and a surface layer

stability parameter, the Monin-Obukhov length.

1

Acknowledgement

I wish to acknowledge the contributions of the reviewers: Dr. Edgar Andreas of

the U.S. Army Cold Regions Research and Engineering Laboratory, Hannover,

NH, and Dr. Donald Hoock and Ms. Gail Vaucher of the U.S. Army Research

Laboratory. In examining this work and providing helpful insights they have

improved both the overall presentation and the chapter 2 integral derivations.

3

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Outer Scale Dened . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Outer Scale versus Mixing Length . . . . . . . . . . . . . . . . . 12

2.3 One- and Three- Dimensional Spectra . . . . . . . . . . . . . . . 13

2.3.1 Atmospheric Refractive Index . . . . . . . . . . . . . . . . 15

2.3.2 Template Three-Dimensional Spectra . . . . . . . . . . . . . 16

2.3.3 The Kaimal Curve . . . . . . . . . . . . . . . . . . . . . 17

2.4 Outer Scale Length . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 The Composite Spectrum . . . . . . . . . . . . . . . . . . . . . 26

3. Turbulent Coherence Diameter . . . . . . . . . . . . . . . . . . . 31

3.1 Kolmogorov Spectrum Results . . . . . . . . . . . . . . . . . . . 32

3.2 Outer Scale Spectrum Results . . . . . . . . . . . . . . . . . . . 32

3.3 Inner Scale Spectrum Results . . . . . . . . . . . . . . . . . . . 35

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Appendix: Integration Methods . . . . . . . . . . . . . . . . . . . . 43

Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5
Figures

1. Fits of m functions to the asymptotic behavior of the proposed curve

of Kaimal et al. [1972]. . . . . . . . . . . . . . . . . . . . . . . . . 18

2. Comparison between Kansas experiment temperature 
uctuation

spectra data and Kaimal et al. [1972] empirical curve. . . . . . . . . . . 18

3. Comparison between the tting method curve involving m functions,

the Kaimal empirical curve, and Kansas data reported by Kaimal et

al. [1972]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4. Outer scale portion of the proposed spectrum compared to the

Kolmogorov and von Karman spectrums. . . . . . . . . . . . . . . . . 23

5. Comparison between the eq. (43) curve and frequency maximum data

reported by Kaimal et al. [1972]. . . . . . . . . . . . . . . . . . . . . 25

6. Comparison between the eq. (43) curve and frequency maximum data

reported by Kaimal et al. [1972] with ordinate axis plotted on a log

scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7. An example of normalized in
uences of inner and outer scale on a

near surface refractive index spectrum. . . . . . . . . . . . . . . . . . 28

8. Outer scale in
uence function for the coherence diameter, plotted with

respect to the parameter Y = Lo=X. . . . . . . . . . . . . . . . . . . 34

9. Inner scale in
uence function for the coherence diameter, plotted with

respect to the parameter Z = `o=X. . . . . . . . . . . . . . . . . . . 35

6
Executive Summary

Synopsis

This report considers the nature of the outer scale of turbulence, its relationship

to the refractive index power spectrum, and the form of this spectrum based on

data reported from the 1968 Kansas experiment [Kaimal et al., 1972].

The outer scale of turbulence is a length scale used to demarcate the beginning

of the so-called inertial subrange. For two points separated by distances greater

than the so-called inner scale length (a few millimeters) and less than the outer

scale length (on the order of meters) 
uctuations in refractive index follow a

\2/3 law". Diculties arise for separation distances outside this range. These

deviations are the sub ject of spectral modications that account for both inner

and outer scale variations.

Under many circumstances the in
uence of the outer scale is not signicant.

For example, optical turbulence eects on beam propagation generally involve

scintillation and beam spread in
uences that are relatively impervious to outer

scale eects due to the limited width of the beam. Imaging of point sources

through turbulence is also little aected by outer scale when considering only

turbulent blurring and scintillation eects since the optics entrance aperture is

smaller than the outer scale.

However, for ob ject jitter and image distortion eects, outer scale in
uences

are likely dominant. The outer scale length has been shown to be directly

related to the spatial frequency that has the greatest in
uence on angle of arrival

variations [Tofsted, 1992]. Yet outer scale is not well dened or quantied.

To correct deciencies regarding outer scale eects characterization, this report

focuses on three issues: (1) Dening outer scale independent of the particular

mathematical form being adopted to describe the refractive index spectrum.

(2) Developing a more accurate refractive index spectral form for low spatial

frequency characterization. (3) Deriving an equation for evaluating outer scale

based on meteorological data measurements.

Results

To dene the outer scale of turbulence requires a knowledge of the form the

refractive index spectrum assumes when inner and outer scale eects are not

considered. This spectrum is called the Kolmogorov spectrum,

nK()=0:033 C2n 11=3; (1)

7
where  is a spatial frequency variable given in units of radians per meter and

C2n is the refractive index structure parameter with units of refractive index

squared (unitless) times meters to the -2/3 power.

The Kolmogorov spectrum is the most frequently encountered spectrum in the

optical turbulence literature because it has the simplest form. This property

makes the Kolmogorov spectrum a useful benchmark, as follows: Assume we

identify some refractive index spectral form which accounts for outer scale eects.

Call it n(). For  values within the so-called inertial subrange we will have

n()  nK(). For  below this range, one enters the energy-containingeddy region in which n < nK, with the ratio n=nK approaching zero as 

approaches zero. Within this framework, the outer scale can be dened as,

n 

1

Lo  =

1

2

nK 

1

Lo 

: (2)

Though arbitrary, heretofore no quantitative denition has been established.

Using this denition, the Kansas 1968 experimental data reported by Kaimal et

al. [1972] were analyzed. While the goal of the analysis was to evaluate the

outer scale length, it was necessary to rst produce a new model of the refractive

index spectrum for low spatial frequencies, . This was necessary because the

Kansas data were plotted as one-dimensional (1-D) spectra (Fn(1)), typical

of sensor data collected through meteorological tower measurements, where the

wind advects spatially 
uctuating air past the tower location. On the other hand,

atmospheric turbulence studies are based on three-dimensional (3-D) spectra.

The only way to compare the two is to propose a 3-D spectrum and transform

it to its 1-D representation, as,

Fn(1)=4 

Z1

0

n q2

1 + 2

r

r dr : (3)

Comparison of spectra led to the development of a relationship between outer

scale and the reported peak frequency of a universal function of scalar property


uctuations in turbulent 
ow. This peak frequency is given in terms of z=L, the dimensionless ratio of height above the surface (z) to the Monin-Obukhov

length (L).

Conclusion

Together, the new denition of outer scale developed, the new spectral form

to describe the eects of the outer scale on the refractive index spectrum, and

the new relationship describing outer scale as a function of height and stability

extend the accuracy of modeling low spatial frequency eects dramatically.

8
1. Introduction

The atmospheric optics community has generally relied on a few select forms for

representing refractive index spectra, as illustrated by the following examples.

Nelson et al. [2000] use a simple Kolmogorov spectrum [Kolmogorov, 1962]

that exhibits 11=3 behavior at all spatial frequencies, where  is a radially

symmetric spatial frequency variable given in units of radians per meter. (This

spectrum was described in the Executive Summary.) Frehlich [2000] augments

the standard Kolmogorov spectrum to account for inner scale eects through

the addition of a Hill [1978] Spectrum term which accounts for the peculiar

bump phenomenon which occurs in the vicinity of the frequency equivalent of

the inner scale length. These studies, and others [Frehlich, 2000; Davis and

Walters, 1994; Martin and Flatte, 1988; etc.], focus on beam wave propagation,

the goal of which is usually to limit, as much as possible, the horizontal spread

of such beams. The narrow cones used in these studies tend to not experience

signicant outer scale eects. And the phase screens used in propagating the

beams are usually centered on the beam path.

Perhaps more sensitive to outer scale eects are studies such as Ellerbroek [1997]

and Young et al. [1998]. The former uses a von Karman spectrum referenced

to Tatarskii [1971]. The latter uses a modied von Karman spectrum which

includes an inner scale roll o region generally attributed to Tatarskii [1971].

Similarly, Yan et al. [2000] used a von Karman spectrum in their adaptive optics

study. Together, the Kolmogorov spectrum, in combination with von Karman,

Tatarskii, and/or Hill spectral modiers, generally span the gamut of

the available choices of spatial frequency spectra for characterizing optical

turbulence. Signicant theoretical and experimental work has been

accomplished in the high spatial frequency regime, with various representations

of this bump eect available [Churnside, 1990; Andrews, 1992; Frehlich, 1992].

All spectra exhibit similar 11=3 dependence for  within the inertial subrange,

and the behavior of the Andrews and Frehlich models are very similar. Of more

concern are dierences between proposed versions of the von Karman spectrum.

While all models of this spectrum are given as,

n() = 0:033 C2n

2 + 2



11=6

; (1)

there is disagreement over the denition of  as well as problems with the

behavior of this functional form at low spatial frequencies.

9
For example, Ishimaru [1978], Ellerbroek [1997] and Young et al. [1998] dene

 as 1=Lo, where Lo is referred to as the outer scale. But Kopeika [1998]

and Cliord [1978] dene  as 2=Lo, yet also refer to Lo as the outer scale.

Obviously two divergent forms of the same equation cannot lead to identical

denitions for the outer scale.

Ishimaru indicated Lo should be on the order of the height above the surface in

appendix C of his book. Kopeika set Lo to kv z, where kv is von Karman's

constant, equal to approximately 0.385 over rough surfaces [Frenzen and

Vogel, 1995], and z is the height above the surface. Combining the uncertainty

about the denition of Lo with the uncertainty about the denition of , we

have over an order of magnitude disagreement between Ishimaru and Kopeika

regarding the positioning of the outer scale cuto.

The von Karman spectrum also has a second problem. At the low frequency

limit, as  ! 0, the spectrum should also approach a zero value. The

von Karman spectrum approaches the constant value 0:033 C2n 11=3

 instead.

The zero limit requirement can be shown through the relationship between the

refractive index spectrum n and the spatial autocorrelation function, n(r)

[Goodman, 1985], given by,n (~r) = hn0 (~x) n0 (~x + ~r)i : (2)

Here, the angle brackets represent an expectation operation, n0 represents

refractive index perturbations at positions ~x and ~x+~r about an average refractive

index value.

n is related to n through a Fourier relationship as will be discussed further

in section 2.3. However, because n is an autocorrelation function of n0 , n must be proportional to the magnitude squared of the Fourier transform of n0

[cf., Gaskill, 1978]. But n0 is a zero mean quantity by denition and hence

(Parseval's theorem) n0's Fourier transform must equal zero at the origin. By

extension, n must also equal zero at the origin. Neither the Kolmogorov nor

the von Karman spectra exhibit this property. Any study purporting to consider optical turbulence eects on image

propagation should begin by considering the outer scale in
uences on the

problem because the outer scale represents the size scale with the greatest


uctuation energy in the spectrum. The available energy normally can be

measured as a function of n() times 42 d, the dierential volume associated

with 
uctuations at a particular spatial frequency scale. Since n is decreasing

as 11=3 across the inertial subrange, the combined eect will be the maximum


uctuations that occur somewhere near the outer scale size.

10
2. Analysis

To address the deciencies in the existing theory discussed in the introduction,

several aspects of optical turbulence related to low spatial frequency phenomena

must be considered. A new form for the refractive index spectrum is proposed

to replace the currently dominant von Karman spectrum. Spectral analysis

arguments are used to indicate the limiting behavior of the spectrum at low

spatial frequency. Analysis of actual data provides further insight into the

detailed behavior of the spectrum in the vicinity of the outer scale frequency, 1=Lo. However, because of ambiguities in denition of the outer scale itself,

it is necessary to clarify the meaning of this quantity rst and then use this

more precise denition in relating the outer scale to measureable meteorological

quantities associated with the atmospheric surface layer.

In section 2.1, a new denition of outer scale is proposed. This new denition

is linked to a property of the outer-scale-in
uenced refractive index spectrum

itself, rather than to a specic form used in describing that spectrum. In

section 2.2, the use of the mixing length is brie
y discussed that has led to some

confusion in the past. In section 2.3, an analysis is performed to link the research

ndings for the Kansas experiment to the three-dimensional (3-D) refractive

index spectrum. The results of this analysis permit both postulating a new

outer scale in
uenced spectrum and establishing a relationship between outer

scale and the Monin-Obukhov length, an atmospheric surface layer stability

parameter. The implications of this functional behavior on the refractive index

itself will then be considered, along with a discussion in section 2.4 of a composite

spectrum that accounts for both inner and outer scale eects.

2.1 Outer Scale Dened

The rst step in describing the outer scale eects should be to uniquely identify

the outer scale in such a way that it is not dependent on the form of the

equation in which it appears. As was discussed in chapter 1, the appearance of

the outer scale in the von Karman spectrum results in an ambiguity based on

dierent literature sources. The same value of outer scale, Lo, has signicantly

dierent ramications when dening  = 1=Lo [Ishimaru, 1978; Andrews

and Phillips, 1998] as opposed to the use of  = 2=Lo [Cliord, 1978;

Kopeika, 1998].

11
To resolve this dilemma I propose the following: Let nK() represent the

Kolmogorov spectrum,

nK ()=0:033 C2n 11=3; (3)

where  is a spatial frequency variable with units of radians per meter and C2n is

the refractive index structure parameter with units of refractive index squared

(unitless) times meters to the -2/3 power. The refractive index spectrum thus

has units of volume. Let n(; Lo) represent any spectrum where outer scale

eects are included, and where the parameter Lo re
ects outer scale eects but

is not necessarily equal to the outer scale. To quantitatively dene the value of

the outer scale length, Lo, we need a new variable,  = 1=Lo. We will dene

the outer scale as that length which satises the relation,

n (; Lo)

nK () =

1

2

: (4)

Typically, Lo appears in functional forms describing n involving only the

product  Lo. Due to the way Lo appears in these functional forms, outer scale

parameters in these equations can always be written as linear relations relative

to the actual outer scale: Lo = Lo. For example, in the case of the von Karman spectrum, when  = 1=Lo is

used, we must have  equal to 26=11  1

1=2

 1:475. Otherwise, using

the denition,  = 2=Lo  is written as, 2

26=11  11=2

 9:269. In

either case Lo maintains the same position relative to the knee of the spectrum

regardless of the form adopted for . 2.2 Outer Scale versus Mixing Length

One hazard to avoid in the available literature is the dierence between outer

scale and a quantity customarily called the mixing length or the characteristic

length. While various mixing length denitions are available, these are

usually employed in providing closure mechanisms for solving the Navier Stokes

equations [e.g., Lewellen, 1977; Zhao, 1994; Finardi et al., 1995]. In this

literature Kopeika's kv z value often appears. For example, Lewellen denes

his characteristic length as,

o = 1:68

u du=dz = 1:68

kvz

m(z=L)

; (5)

where u is the friction velocity, du=dz is the vertical gradient of the horizontal

windspeed, and m is a diabatic in
uence function of momentum which is a

dimensionless function of the ratio of the height (z) divided by the MoninObukhov length (L). I previously used this parameter for dening outer scale

[Tofsted, 1992], though in hindsight it appears inappropriate. Zhao [1994],

12
and Finardi et al. [1995], use a form of the mixing length scale developed by

Blackadar [1962]:

1

o =

1

1

+

1

kvz

; (6)

where 1 is the asymptotic value of o at large height. However, in no instance

is this result linked to a turbulence spectrum. Hence, applications requiring an

outer scale should look to other sources to determine this quantity. Such a source seems to be exclusively available through data illustrated in

or derived from Kaimal et al. [1972]. Nevertheless, use of this information is

complicated by the mode of presentation of the data. Optical turbulence work

usually involves integration of 3-D spectra, but the Kansas experimental data

presented by Kaimal involves one-dimensional (1-D) spectra. Hence, to use the

Kaimal data a turbulence spectrum must be posited, and then this spectrum

must be converted to a 1-D representation. This task is treated in the following

section. This reference can also be found along with many of the other articles

cited in this report in the SPIE Milestone series compilation - Turbulence in a

Refractive Medium, Andreas [1990].

2.3 One- and Three-Dimensional Spectra

Within the optical turbulence research community the assumption of 3-D

homogeneous and isotropic turbulence is typically made. This assumption is

useful, though in most cases it is false. Hinze [1987] points out that homogeneous

turbulence is only possible under decaying turbulence conditions. To ensure

that the turbulence is maintained requires nonhomogeneous energy sources.

Nevertheless, under most conditions the source terms generating additional

turbulence are small. I thus adopt a locally homogeneous condition.

Theory also suggests that turbulence is not isotropic at  < 1=Lo. However,

again, while use of the isotropic assumption is not entirely appropriate, the

largest deviations occur only at the lowest frequencies, and these lowest

frequencies have the least eect on the optical properties normally studied.

Therefore, from a tractability view and due to its minimal impact, an isotropic

turbulence assumption will be adopted throughout this discussion. In general,

though, we do not need to adopt a homogeneous turbulence assumption in

order to study properties of the refractive index power spectrum. Further, it is

generally assumed that turbulence has a vertically inhomogeneous structure.

With these caveats made, I proceed with the general development, rst

considering the limitations of the von Karman spectrum.

As was mentioned in the introduction, the refractive index spectrum is related

to the autocorrelation function, n(r). Or, more generally, for scalar property

13
, the relationship between its autocorrelation function and its spectrum may

be written as [Beland, 1993],

 (~) = 

1

2  3 ZZ1Z

1

 (~r) exp (i ~  ~r) d~r; (7)

where ~ = (1; 2; 3), ~r = (x; y; z), and i = p

1.

The Fourier relation between  and  permits the writing of the inverse

transform relation,

 (~r) = ZZ1Z

1

 (~) exp (i ~  ~r) d~; (8)

When the assumption of isotropy is invoked these equations may be reexpressed

in spherical coordinates and integrated over their spherical angular coordinates

rst, leaving functions dependent only on the magnitudes  = j~j and r = j~rj.  () = 

1

2  3 Z1

0

 (r)

sin( r)

 r

4  r2 dr; (9)

 (r) = Z1

0

 ()

sin( r)

 r

4  2 d: (10)

Most atmospheric optics calculations are made in terms of weighted integrals

involving n. But the refractive index is a function of temperature, pressure,

and humidity 
uctuations in the atmosphere, with temperature 
uctuations

being the dominant eect for visible to infrared frequencies. To measure

these 
uctuations directly usually involves tower measurements which record

the temporally 
uctuating properties of the atmosphere as the wind advects

positionally varying air through the sensor mast location. Taylor's [1938] frozen

turbulence hypothesis is then used to relate the temporal 
uctuations observed

to spatial 
uctuations through extrapolation based on the mean wind speed.

The temporal records of the tower data can be analyzed, but these data only

represent 1-D spectra. These spectra are related to their 3-D counterparts

through the relation [Panofsky and Dutton, 1984],

F (1)=2 Z1

0

 q

2

1 + 2

r

2r dr ; (11)

where r is a polar spatial frequency coordinate in the (2; 3) plane,

corresponding to spatial coordinates normal to the wind direction. Note also

that F is only dened for positive 1 values. Due to  function symmetry

about zero, results at negative 1 values are folded onto the positive portion

of the axis. This approach leads to the factor of 2 in front of the integral in

eq. (11).

14
2.3.1 Atmospheric Refractive Index

I now note that although  was used to denote an arbitrary scalar property of

the atmosphere, a so-called conservative passive additive, our real interest is in


uctuations of the atmospheric refractive index. Optical turbulence is caused

by 
uctuations in this index about its mean value. The mean refractive index

is a function of air temperature, pressure, and humidity. Positional changes in

these properties alter the refractive index of the atmosphere (n).

In fact, n is so close to unity in the atmosphere that a secondary variable, N, the refractivity, is normally used to describe n's variability:

n =1+ N 106: (12)

The functional dependence of N is given in terms of its dry air behavior in

combination with water vapor eects [Hill et al., 1980],

N = Ndry + Nvapor ; (13)

Ndry = 0:3789 

1 + (5:337  0:0157 T ) Pt 106 Pt N0

T

; (14)

where Pt is the air pressure in torr, and T is the air temperature in Kelvin (K).

N0 expresses the wavelength dependence, given by, N0 = 64:328 +

29498:1

(146  2) +

255:4

(41  2)

; (15)

where  (m) is the radiation wavelength in the visible through far infrared

band.

The water vapor in
uence on the refractive index is given by, Nvapor = 0:0059 

1  0:0109 2 A; (16)

where A is the absolute humidity (g/m3).

At visible through far infrared wavelengths, although there are humidity and

pressure dependencies, temperature variations tend to clearly dominate the


uctuations in the refractive index about its mean. A secondary correlation

between temperature and humidity 
uctuations has been characterized by

Wesely [1976], though the main correlation eect can be accounted for through

use of the Bowen ratio [Kunkel and Walters, 1983]. These temperature


uctuations are driven largely by the energy 
ux of sensible heat at the earth's

surface [Oke, 1978].

To quantify the variations in refractive index due to temperature, we write the

dierential of n with respect to temperature,

dn = dN 106  78:2 106 P

T 2 dT ; (17)

15
where pressure, P , has been converted to units of millibars (mbar) (1 torr =

1.33322 mbar). The important aspect to be emphasized here is that 
uctuations

in refractive index are directly proportional to 
uctuations in temperature. We

can thus equivalently study either a turbulent eld of temperature 
uctuations

or refractive index 
uctuations, and we will be studying the same phenomenon.

To directly study these 
uctuations, the refractive index is normally rewritten

as an average quantity, n0, and a 
uctuation component, n0. n = n0 + n0: (18)

There are no optical turbulence eects due to n0, while the n0 statistic was

previously discussed in connection with the autocorrelation function, n(r).

2.3.2 Template Three-Dimensional Spectra

Hence temperature and refractive index 
uctuations are strongly coupled, and

once the structure of the temperature spectrum is understood, the refractive

index spectrum immediately follows, having the same form. The diculty

remains, however, in determining an appropriate 3-D spectrum when only 1-

D measurement data are available.

To solve this problem a family of template 3-D spectra will be used. These can

be transformed into 1-D spectra and analyzed with respect to available empirical

curves and data reported in the literature. This family of functions has the form,

m (; L) = L11=3

( L)2m

h1+( L)2i(11+6m)=6

; (19)

such that for   L, m(; L) ! 11=3. Based on this family, the standard

von Karman spectrum [Tatarskii, 1971] of turbulence can be written, using the

m = 0 member of this family, as,

nV () = n C2n 0 (; Lo) : (20)

where n is an exact form replacing the use of 0.033,

n =

5

36

22=3 (5=6)

3=2 (2=3)  0:033: (21)

In defense of Theodore von Karman, I would note that it is arguable that he

did not actually propose the spectrum honoring his name. von Karman was

certainly aware that the limiting behavior of turbulent spectra at low frequencies

approaches a zero value at the origin. Hinze [1987] attributes the equivalent of

1 to von Karman [1948], while he attributes the use of 0 to Saman [1967].

All members of this family behave similarly at high spatial frequency , where

they exhibit 11=3 dependence. But at low spatial frequencies 0 approaches

a nonzero constant.

16
2.3.3 The Kaimal Curve

To study the behavior of this family in comparison to available data, the

m functions were rst transformed to 1-D spectra using eq. (11). These 1-

D functions were then multiplied by frequency variable 1 so they could be

compared to a `universal' empirical curve suggested by Kaimal et al. [1972].

This equation was most recently expressed by Kaimal and Finnegan [1994] in

the form,

f S (f )

2

 =

B (f =f0)

1 + B (f =f0)5=3

; (22)

where  is the property under study (temperature, humidity, etc.), 2



is the

variance of this property, S (f ) represents the 1-D spectral content of the


uctuating portion of this property expressed in terms of a unitless temporal

frequency variable f , B is an empirically t constant, and the parameter f0 is

related to the peak frequency, fm. In their original paper, Kaimal et al. [1972]

found B = 0:16. More recently, Kaimal and Finnegan [1994] report B = 0:164.

This function is termed universal because Kaimal et al. viewed it as applicable

to any scalar property. Hereafter, I shall refer to eq. (22) simply as the Kaimal

curve.

In order to show the appropriateness of members of the template family in

approximating the Kaimal curve, the m function 1-D versions were tted to

the asymptotic behaviors of the Kaimal curve at both high and low spatial

frequency. The method of tting involved shifting the transformed template

function vertically through multiplication by a constant and horizontal shifting

by multiplying the frequency variable by a scaling multiplier.

As an example, 1, when converted to the Kaimal form, is expressed as,

	1(f

0) = 0:6 f

0 f02 + 6=11

(1 + f 02)11=6

: (23)

This form is tted to the asymptotic behavior using 1:02045 	1(0:47909 f ). The

resulting tted curves are plotted against the Kaimal curve in gure 1 for cases

m = 0 to 3. As seen, the m functions exhibit peaks which are progressively

narrower than the Kaimal curve.

In order to approximate the Kaimal curve using these template functions, it was

realized that two of these functions could be combined in such a way that the

asymptotic behavior remained the same, but the peak region could be better

approximated.

Due to its incorrect limiting behavior at low frequency (it approaches a constant

rather than zero) the 0 function was not used. Having ruled out use of the 0

function, the next two members of the  family were used:

 / A 1 (; La) + (1  A) 2 (; Lb) ; (24)

17
Figure 1. Fits of m functions to the asymptotic behavior of the proposed curve

of Kaimal et al. [1972].

Figure 2. Comparison between Kansas experiment temperature 
uctuation

spectra data and Kaimal et al. [1972] empirical curve.

Figure 2 shows the data used in this analysis along with an empirical curve t

to this data as contained in gure 20 of Kaimal et al. [1972].

Mathematically, we can relate this comparison to a new form for the outer scale

18
in
uenced 3-D refractive index spectrum,

n() = n C2n 2

6

4

A1 ( La)2 L

11=3 a h1+( La)2i17=6 +

(1  A1) ( Lb)4 L

11=3

b

h1+( Lb)2i23=6

3

7

5 : (25)

The La and Lb parameters appear as placeholders which are related to the peak

frequency (fm) of Kaimal et al. [1972] and have a direct relationship to the outer

scale, but the exact nature of this relationship will only be obtained as a result

of the analysis.

Using this form for the refractive index spectrum we may write the variance of

refraction index 
uctuations as,

2n =

Z1

0

n() 4 2 d =

12

5 n C2n AA

A1 

L2=3 a 

15

17

L

2=3

b  +

15

17

L

2=3

b 

;

(26)

where AA is an integration constant (see appendix),

AA =

3=2 33=2

11 

5

6  



2

3  : (27)

The 1-D spectrum for the refractive index can be generated from the 3-D

spectrum through another integration (see appendix).

Fn (1)=2 Z1

0

n q2

1 + 2

r

2 r dr

= A1

12

5 n C2n L5=3 a h

(1 La)2 + 6

11 i

h1+(1 La)2i11=6

+ (1  A1) 12

5 n C2n L

5=3

b

h

(1 Lb)4 + 12

11 (1 Lb)2 + 72187 i

h1+(1 Lb)2i17=6

:

(28)

Spherical symmetry of the spectrum, n, is assumed such that the integral

results at negative 1 can be combined with positive 1 results, producing a

one-sided spectrum containing only nonnegative values of 1. Hence, the factor

2 appears in front of the integral sign.

We now combine the results of the previous calculations to produce the

expression 	n(1) = 1 Fn(1)=2n. Due to the normalization by 2n quantities

19
n and C2n divide out. The remaining expression is independent of the scalar

variable being studied.

	n(1)

1 =

L

5=3 aAX

h

(1 La)2 + 6

11 i

h1+(1 La)2i11=6

+

L

5=3

b (1  A1)

A1 AX

h

(1 Lb)4 + 12

11 (1 Lb)2 + 72187 i

h1+(1 Lb)2i17=6

; (29)

where,

AX = AA

L2=3 a +

15

17 

1

A1  1

L

2=3

b 

(30)

This expression is written in terms of the variable 1, while the Kaimal equation

is written in terms of the variable f . 1 is a spatial frequency representation

along the x axis which is normally understood as the direction of mean wind


ow in meteorological studies. The f variable is a dimensionless temporal

frequency variable related to tower sensed meteorological data. The two are

related through the equation, 1 z = 2 f , where z is the height above the

surface. To permit comparison let us dene the variable 0 analogous to f0, such that 0 = 2 f0=z. Eq. (22) may therefore be rewritten in terms of the

dimensionless frequency ratio 1=0 = f =f0, 	n(1)

1



B=0

1 + B (1=0)5=3

: (31)

The behavior of the 	 (1) =1 function must thus obey B=0 dependence at

low frequency and 5=3

1 

2=3

0 dependence at high frequency. Considering the high frequency behavior, cancellation results in an expression

for 0 as a function of A1 and AX :



2=3

0 =

1

A1 AX

: (32)

Rewriting this expression, we have the condition:

1

AA = A1 (La0)2=3 +

15

17

(1  A1) (Lb0)2=3: (33)

At the low frequency limit we obtain the expression,

B

0 =

h

6

11 A1 L

5=3 a + 72

187 (1  A1) L

5=3

b i

A1 AX

: (34)

20
The denominator on the right is just 2=3

0 . Eq. (34) then becomes,

11

6

B = A1 (La 0)5=3 +

12

17

(1  A1) (Lb 0)5=3: (35)

We thus have two equations in four variables: A1, La, Lb, and 0. However,

because of the relationships between La, Lb, and  in eq. (25), and the denition

given for the outer scale in eq. (4), we presume that La, Lb must be proportional

to the outer scale length Lo. 0 appears to perform a similar role in Kaimal's

equation, except that it must be inversely proportional to Lo. Hence, the

products La 0 and Lb 0 may be viewed as dimensionless functions which

eectively displace their respective  functions along the  axis. We then

have only three unknowns (
a = La 0, 
b = Lb 0, and A1) in two equations.

Conceptually, we might view A1 as a free parameter which then denes 
a and


b. But there is a complication that arises when solving for an optimal value of A1. From gure 1 it is observed that both 1 and 2 lie above the Kaimal curve.

This means that the value of A1 may not lie between 0 and 1 since what we wish

to accomplish is to subtract a multiple of the upper curve (2) from a multiple

of the lower curve (1), eectively pulling the 1 line downward in the center

toward the Kaimal curve and the Kansas data. This is only possible if A1 is

greater than unity. Unfortunately, when A1 > 1 there is no general solution to

both of these equations simultaneously for a general B value.

The best that can be hoped for is to try to approximately match the slope

behavior given in eqs. (34) and (35), while choosing a value for A1 that provides

a good t to either the available data or the Kaimal curve. Originally, I had

conceived that it would be best to t to the Kaimal curve, but this did not

seem reasonable, particularly in view of the Kaimal curve's performance at low

frequency. There, Kaimal et al. [1972] recommended a value for B of 0.16. Later,

Kaimal and Finnegan [1994] amended this constant to B = 0:164. However,

examining the data it appears that an asymptotic constant of B = 0:20 is a

better t.

In any case, the dispersion of the data is signicantly larger at low frequencies

than it is at high frequencies. It therefore makes sense that if only one curve

can be solved for it should be the high frequency equation. Therefore, solving

for 
b in terms of A1 and 
a, we have,


b =

2

4

17

15



1

AA  A1 


2=3 a 

(1  A1)

3

5

3=2

: (36)

For the low frequency curve we have,




0b =

2

4

17

12



11

6 B  A1 


5=3 a 

(1  A1)

3

5

3=5

: (37)

21
Of interest is that we may choose a value for A1 and then vary 
a and look at

a plot of the function 
b(A1; 
a)  
0b(A1; 
a) to see how this function varies.

This dierence approaches zero for a narrow range of 
a values regardless of

the value of A1. Thus, although there is no solution to eqs. (34) and (35) in

general, we may approach close to a solution and use this result. A relatively

good choice of parameters that appears to come close to both the centroid of

the data and t for both high and low frequency limiting behaviors is A1 = 8:2,


a = 0:44, 
b = 0:5254. The 
b result compares well with 
0b = 0:5221, though

it is not exactly equal. Figure 3 shows the resulting curve comparison with the

Kansas data. As can be seen, the low frequency behavior of the proposed curve

appears to t to the centroid of the Kansas data better than the Kaimal curve

of gure 2.

Figure 3. Comparison between the tting method curve involving m functions,

the Kaimal empirical curve, and Kansas data reported by Kaimal et al. [1972].

With these results, we may return to the description of the proposed 3-D

refractive index spectrum in eq. (25). This equation can be rewritten in terms

of n, C2n, 0, 
a, and 
b:

n() = n C2n 11=3

0

2

6

4

A1 ( 
a=0)2


11=3 a h1+( 
a=0)2i17=6 +

(1  A1) ( 
b=0)4


11=3

b

h1+( 
b=0)2i23=6

3

7

5 :

(38)

With A1, 
a, and 
b xed, n a constant, and C2n constant for any given moment

and location, the remaining parameters are  and 0. Using the denition of

22
the outer scale in eq. (4), we nd for the parameter values listed above,



0

 4:71393; 0 

0:21214

Lo

: (39)

Writing the same result in terms of La and Lb, 0 La = 0:4400 ! La = 2:0741 Lo; (40)

0 Lb = 0:5254 ! Lb = 2:4767 Lo: (41)

The resulting outer scale spectrum seen in gure 3 accurately models the

refractive index spectrum to spectral frequencies well below the knee associated

with  (by approximately a factor of 5), thus signicantly extending the

simulation of the refractive index spectrum to very low wavenumbers. A plot of

the outer scale region of the resulting 3-D spectrum is seen in gure 4.

Figure 4. Outer scale portion of the proposed spectrum compared to the

Kolmogorov and von Karman spectrums.

In this gure the newly developed spectrum is compared with both the

Kolmogorov and von Karman spectrums. While the von Karman spectrum rolls

o to a constant value at low frequencies, the Kolmogorov spectrum increases

to a singularity at zero frequency while the new spectrum rolls over toward a

zero value at the origin. Of interest is a comparison between the von Karman

spectrum and the proposed spectrum. The von Karman spectrum reaches a

maximum at a frequency slightly below the outer scale frequency  while the

proposed spectrum continues to increase to a peak approximately twice that of

the von Karman spectrum.

23
2.4 Outer Scale Length

We are now in a position to discuss the association between the outer scale length

and the frequency maximum measured during the 1968 Kansas experiment.

In the previous section, we connected the outer scale length, Lo, to the 0

parameter. In this section, 0 is related to frequency maximum data collected

during the 1968 Kansas experiment. This connection leads to a denition of

outer scale as a function of similarity parameters in the atmospheric surface

layer.

To build this connection, note that 0 is dened as 0 = 2 f0=z. But f0

is related to the frequency maximum, fm, at which Kaimal's universal scalar

curve peaks. This curve is given by,

F (f ) = B (f =f0)

1 + B (f =f0)5=3 : (42)

The peak of this curve is found by setting its derivative to zero. The location of

the peak is denoted by fm. Solving for this peak location we nd,

f0 = fm 

2B

3 3=5

: (43)

For the original B value (0.160), f0 = 0:2611 fm. For the revised value, B =

0:164, f0 = 0:2650 fm. The two are relatively close, and since the Kaimal et

al. [1972] analysis was based on the 0.160 value, we will use this result.

We have thus logically connected the outer scale value Lo to the frequency

maximum location fm. We nd a functional form for fm through an analysis of

Kaimal et al.'s gure 18. The consolidated data from this gure are included in

both gures 5 and 6 in this document. In these gures, the abscissa coordinate

is the ratio z=L, representing the ratio of height above the surface (z) to the

similarity parameter L known as the Monin-Obukhov length. The rst of these

gures (gure 5) presents fm in a linear plot format as a function of z=L. The

second gure (gure 6) presents fm in log form as a function of z=L. Along

with the data are two curves, one for L < 0 and one for L > 0. The distinction

between these conditions is termed the stability:

fm =



0:460 p

+ zL + 0:0014 zL > 0;

0:036 p zL + 0:0014 zL < 0. (44)

Negative L values correspond to daytime (unstable) conditions, where parcels of

air displaced vertically tend to accelerate away from their level of origin. Positive

24
Figure 5. Comparison between the eq. (44) curve and frequency maximum data

reported by Kaimal et al. [1972].

Figure 6. Comparison between the eq. (44) curve and frequency maximum data

reported by Kaimal et al. [1972] with ordinate axis plotted on a log scale.

L values indicate stable conditions where air parcels tend to return to their level

of origin.

Although Kaimal et al. [1972] included a hand drawn curve t to the data, the

25
above relationship is also a reasonable t. This functional relationship reveals

several interesting features regarding outer scale dependencies within the surface

layer atmosphere. First, it reveals that there are two pertinent length scales that

aect the outer scale length. Focussing on the results for B = 0:164, we nd,

for nonnegligible heights, z,

Lo 



0:281 p

+z L L > 0, jz=Lj > 0:1;

3:592 pz L L < 0, jz=Lj > 0:1. (45)

This functional relationship not only reveals that the outer scale depends on both

the height and the Monin-Obukhov length, it also reveals that the dependence

is equal between these two in
uencing variables. However, the outer scale

exhibits considerably longer lengths under unstable atmospheric conditions than

under stable conditions. Though this dierence is likely due to bouyancy

dierences between the unstable daytime and the stable nocturnal surface layer

atmospheres, the exact mechanism responsible for these dierences cannot be

discerned from this study. Writing the full form of the fm function,

Lo = 8

<

:

0:281 z p+z=L + 0:0030z=L > 0;

3:592 z p

z=L + 0:0389z=L < 0.

(46)

This form reveals explicitly how Lo is dependent on the Monin-Obukhov length.

Both C2n [cf., Rachele and Tunick, 1994] and the inner scale length, `o [cf.,

Tofsted and Auvermann, 1991] are also known to vary with atmospheric stability

conditions. Hence, modeling atmospheric surface layer conditions will produce

the parameters needed to predict the form of the refractive index spectrum as

a function of height.

2.5 The Composite Spectrum

An application involving this new spectrum is as part of a calculation of

the turbulent coherence diameter (chapter 3). Outer scale in
uences on the

coherence diameter can be signicant under certain circumstances. However, to

evaluate the coherence diameter it is imperative that inner scale eects also be

included. Therefore, it behooves one to modify the refractive index spectrum to

include both inner and outer scale eects. Andrews and Phillips [1998] present

several inner scale simulation functions, including their modied spectrum, the

Tatarskii [1971] spectrum, and the Hill spectrum [Hill and Cliord, 1978].

These spectra will not be considered here in detail. Rather, I note that they

typically either ignore outer scale eects altogether or simply replace the 11=3

Kolmogorov dependence with a term associated with the von Karman spectrum,

2 + L211=6

. Inner scale eects are either represented as a series of terms as

26
in the Andrews and Phillips modied spectrum or as a superimposed multiplying

factor riding on the main Kolmogorov dependence.

In either case, it is dicult to analytically evaluate the eects of both inner

and outer scales because once the spectrum is represented as a single functional

form it becomes dicult to integrate. Hence, I introduce here the composite

spectrum. This new spectrum is composed of three parts. The rst two of these

parts represent a summation of two spectra, one representing only inner scale

eects, the other only outer scale eects. The high frequency portion of the

outer scale spectrum and the low frequency portion of the inner scale spectrum

both re
ect Kolmogorov behavior. The summation of these two spectra must

therefore be balanced by the subtraction of a Kolmogorov spectrum from the

sum. The spectrum thus produced includes both inner and outer scale eects

while simultaneously permitting separate integration of the individual parts,

avoiding, as much as possible, computational complexity. This spectrum is written as:

n(; `o; Lo)=nO(; Lo)+nI (; `o)  nK(); (47)

where nK() is again the Kolmogorov spectrum (eq. (3)), the outer scale

modied spectrum, nO, is synonymous with eq. (25), and

nI () = n C2n 11=3 FF V 

 `o 7:4  ; (48)

is an approximation [Tofsted, 1991] to Frehlich's [2000] inner scale modied

spectrum based on an analysis of measurement data, where,

V (x) = log10(x)+0:72; (49)

FF (V ) =

8

><

>:

1; V < 3;

1+0:424 exp 2:725 V 2; 3  V < 0;

0:712 1 + cos  V

0:9  1  0:786 V 2 0  V < 0:9;

0 0:9  V . (50)

Although the value 7.4 appears in eq. (48) above as a constant, this quantity

is actually somewhat variable. The ratio `o=7:4 is supposed to represent the

Kolmogorov microscale , but the ratio between `o and  depends on the Prandtl

number, which is itself temperature dependent [Hill, 1997].

An example of the behavior of the combined eects of inner and outer scale are

seen in gure 7. In this gure, the spectrum has been normalized by dividing

by the Kolmogorov spectrum. The inertial subrange portion of the spectrum

thus evaluates to a constant value of unity in the curve's central section. The

inner scale bump section is shown on the right hand portion of the curve and

the outer scale section is seen on the left.

27
Though an outer scale value of 5 m was used, this value is not necessarily typical.

Given the variability of outer scale on height and the Monin-Obukhov length,

it is dicult to speak in generalities, yet, as is predicted by eq. (46), we should

generally expect the width of the inertial subrange to be approximately an order

of magnitude wider under daytime conditions than at night due to the functional

dierence in eq. (45) between stable and unstable conditions. Regarding the

inner scale, gure 7 highlights the positions of two spatial frequencies related

to the inner scale, `o, and the Kolmogorov microscale, . The two are related

through the equation,  = `o=7:4 [Hill and Cliord, 1978], which also explains

the appearance of the factor 7.4 in the argument to the V function of inner scale

adjustment factor FF .

Figure 7. An example of normalized in
uences of inner and outer scale on a

near surface refractive index spectrum.

As seen in the gure, the eects of the outer scale extend for approximately

one-and-a-half decades. Inner scale eects have a similar extent. Hence, even

though the inner-to-outer scale ratio is approximately a factor of 1000, there is

still only about a decade and a half associated with the inertial subrange for this

particular choice of inner and outer scale values. The eects of inner and outer

scale are thus on the order of signicance of the inertial subrange in this case.

The question may be asked whether this situation is typical. Studies of

atmospheric eects on propagation often ignore inner and/or outer scale regions.

The rationale for avoiding consideration of these eects most often concerns ease

of computation. In the case of the proposed outer scale spectrum we have a form

which is not necessarily easy to use, but which is nevertheless not intractable.

The main question is whether such calculations are necessary and the answer is

28
likely that under most conditions the answer is no, for the simple reason that

the outer scale may be so large (because of atmospheric conditions or the fact

that the line of sight is suciently removed above the surface) that other factors,

such as the weighting function used in performing integrations with respect to

frequency, make the resulting calculations insensitive to spectral dependence on

outer scale. Nevertheless, there will be certain conditions involving surface-tosurface propagation where both inner and outer scale in
uences on the optical

turbulence parameterizations will be signicant.

29

3. Turbulent Coherence Diameter

As an application, consider evaluation of the coherence diameter using the

composite spectrum. The coherence diameter is given implicitly in the equation

[Hill and Cliord, 1981],

1=42 k

2 Z

S

0

ds Z1

0

h1  J0 

sro 2:1 S

i n(; Lo(s); C2n(s); `o(s))  d; (51)

where Lo, `o, and C2n can all be functions of path position. In this equation, k

is the radiation wavenumber (2=), s is a path position variable, S is the total

path length, and ro is the turbulent coherence diameter, often called the Fried

coherence diameter [Fried, 1966].

Path position dependent parameters are most generally considered height

dependent. Since the terrain underlying the path between ob ject and observer

is likely to be non
at and/or a slant path, the values of these quantities will vary

in nonlinear fashion. However, at a given moment for a given point along the

optical path, the governing parameters can be considered constant. It is thus

possible to directly evaluate the inner integral.

To evaluate this equation we begin by introducing the new length variable,

X = s ro=(2:1 S). Using the composite spectrum introduced in the previous

chapter, we may write eq. (51) as,

1=4 2 k

2 S

o Z

o

0

n C2n(XS=o) X5=3[G(X; Lo) + H(X; `o)  @] dX; (52)

where the path integration variable has been replaced by X, o = ro=2:1,

and G, H, and unity represent separate results for dierent components of

the turbulence spectrum. G represents the combined outer scale eects, H

represents the combined inner scale eects, subtraction by the constant @

represents the Kolmogorov eect, and Lo and `o may vary as a function of

range.

To solve for ro involves an iterative method which requires an initial value of ro to

begin the search procedure. In general, ro will be greater than this initial value,

which is normally computed based on a Kolmogorov spectrum assumption.

31
3.1 Kolmogorov Spectrum Results

For the Kolmogorov portion of the spectrum, the inner integral may be written,

n C2n Z1

0

[1  J0 ( X)] 11=3  d  0:033 1:11833 C2n X5=3: (53)

The constant terms then combine as, 42 0:033 1:11833  1:45695. Through

further analysis, we nd that the integration constant, 1.11833, is actually the

approximation of the expression,

@ =



25=3 (11=6) (11=6)  1:118334400: (54)

Returning to the main double integral in eq. (51), we have,

1=42 k

2 n @

Z

S

0

C2n 

s ro 2:1 S

5=3

ds; (55)

which can be solved explicitly for ro as,

ro = 2:1 2

4

42 n @ k

2 S

Z

1

0

C2n(S w) w5=3 dw

3

5

3=5

; (56)

where w = s=S is a dimensionless path position variable and C2n may vary as

a function of range. This form corresponds to typical integral representations

of the coherence diameter for the Kolmogorov refractive index spectrum [cf.,

Roddier, 1981].

One important factor to consider in the use of this equation regards the

associated path position weighting function, w5=3. In imaging problems, this

weighting assumes the imager is located at range S (w = 1). In propagation

problems (laser beams), this weighting function is normally reversed such that

the laser source is located at the origin. In this case, a weighting function of

(1  w)5=3is used.

3.2 Outer Scale Spectrum Results

To evaluate the outer scale cases we now introduce three new dimensionless

variables, Ya = La=X, Yb = Lb=X, and u =  X. Given the spectral form

32
associated with the outer scale, evaluation of the inner integral in eq. (51) will

involve the solving of equations having the form,

Gm(L=X) = X5=3 Z1

0

[1  J0( X)] m(; L)  d

= Gm(Y ) = Y 11=3 Z1

0

[1  J0(u)] (uY )2m

[1 + (uY )2]m+11=6 u du: (57)

Thus, the factor X5=3sifts out of these equations just as it does from the

Kolmogorov expression.

After considerable math, it can be shown that,

G1 (Ya) = Y 11=3 a Z1

0

[1  J0(u)] (uYa)2

[1 + (uYa)2]17=6 u du =

18

55

Y 5=3 a

+

18

55

21=6





5

6  Y

1=6 a 

1

2

K1=6 

1

Ya   Ya K5=6 

1

Ya  : (58)

It can also be shown that in the limit of large Ya,

G1 (Ya) = Y 11=3 a Z1

0

[1  J0(u)] (uYa)2

[1 + (uYa)2]17=6 u du @ 54

55

Y 1=3 a : (59)

For the second function we nd the following results:

G2 (Yb) = Y

11=3

b Z1

0

[1  J0(u)] (uYb)4

[1+(uYb)2]23=6 u du =

216

935

Y

5=3

b

+

216

935

21=6





5

6  Y

1=6

b 

23

24

K1=6 

1

Yb  



Yb +

1

8 Yb  K5=6 

1

Yb  : (60)

In the limit of large Yb for this function we nd,

G2 (Yb) = Y

11=3

b Z1

0

[1  J0(u)] (uYb)4

[1 + (uYb)2]23=6 u du @ 972

935

Y 1=3

b

: (61)

In these results, the functions K (x) are modied Bessel functions of the second

kind, order  [cf., Kreyszig, 1972]. However, the raw integration results are

33
expressed in terms of modied Bessel functions of the rst kind, I (x), which

are related to functions of the second kind through the relation,

K (x) = 

2 sin()

[I (x)  I (x)] : (62)

These calculations also involve use of the recurrence relationship,

I (x) = I2(x) 

2

x

I1(x); (63)

and the series expansion form of I (x),

I (x) = X1

m=0

x2m+ 22m+ m! (m +  + 1) : (64)

Combining these eects, we generate the function,

G(Ya; Yb) = A1 G1(Ya) + (1  A1) G2(Yb): (65)

Let Y = Lo=X. At large Y we have,

G(Y ) @ 0:78078 Y 1=3: (66)

The general solution is plotted in gure 8. In this gure the approximation

function was used for Y values greater than 100 due to numerical stability

problems encountered in evaluating the Bessel functions at high argument values.

The approximation at this level is accurate to 5 digits of accuracy.

Figure 8. Outer scale in
uence function for the coherence diameter, plotted with

respect to the parameter Y = Lo=X.

34
3.3 Inner Scale Spectrum Results

The results for the inner scale eects must generally be evaluated numerically

due to the form used in describing the inner scale modier to the spectrum. But

the form of this calculation is essentially similar to the one made for the outer

scale,

H(`o=X) = X5=3 Z1

0

[1  J0( X)] 8=3 FFV

`o 7:4  d (67)

This calculation can be written as a function of the dimensionless variable Z =

`o=X. The results of these calculations are shown in gure 9.

Figure 9. Inner scale in
uence function for the coherence diameter, plotted with

respect to the parameter Z = `o=X.

To study the behavior of this function at large Z we perform the change of

variables u = X,

H(Z) =

11:4=Z

Z

0

[1  J0(u)] u8=3 FFV

u Z

7:4  du: (68)

The upper limit on this integration is due to the upper cuto of the bump

spectrum function FF combined with the denition of the V function. For large

Z the limits of the integration will become small, to the point that the quantity

[1J0(u)] u8=3 can be approximated by u2=3=4, using only the rst two terms

of the Bessel function expansion.

35
Considering the behavior of this portion of the integrand, and considering that

the FF function is almost constant throughout its nonzero range, it should

perhaps come as no surprise that the integral evaluates approximately as 3u1=3=4

with an upper limit proportional to 1=Z. Analysis of calculated integration

results for 101 < Z < 103 yields a functional form of approximately, H(Z)  0:61068 Z1=3

; Z  1: (69)

This result is used to evaluate the H(Z) at Z values where numerical techniques

fail due to integration diculties near the singularity at u = 0.

36
4. Conclusions

In this report, the modeling of outer scale eects on optical turbulence within

the atmospheric surface layer has been considered. We rst discussed the means

used in treating outer scale eects by previous investigators. We considered

prior limitations in the denition of outer scale and in the use of the standard

von Karman spectrum. A new denition of outer scale was given such that

a particular point in the energy-containing-eddy region of the refractive index

spectrum is used to denote the outer scale wavenumber, . A development was

also described which relates outer scale eects on the refractive index spectrum

to the research results of Kaimal et al. [1972] and a subsequent reanalysis

by Kaimal and Finnegan [1994]. These results link the outer scale value

to frequency maximum data for temperature 
uctuations in the surface layer

atmosphere. Analysis of this data and the subsequently proposed dimensionless

height (z=L) parameterization of the frequency maximum (fm) indicates that

the outer scale is not merely a function of height above the surface, as has

been commonly assumed, but rather is a function of height and the MoninObukhov length, rmly establishing that the outer scale is a function of both

of the length parameters associated with the surface layer. As part of this

analysis, a new outer scale in
uenced spectrum was developed which avoids

the zero limit problem of the von Karman spectrum. Lastly, the implications

of the outer scale were considered in combination with inner scale eects

within a proposed composite refractive index spectrum. The resulting spectrum

permits the separable evaluation of inner and outer scale eects on atmospheric

propagation. As an application, we considered the evaluation of the turbulent

coherence diameter based on this new spectral model.

37

References

Andreas, E.L., ed., 1990, Selected Papers on Turbulence in a Refractive Medium, SPIE Optical Engineering Press, Bellingham, WA.

Andrews, L.C., and R.L. Phillips, 1998, Laser Beam Propagation through

Random Media, SPIE Optical Engineering Press, Bellingham, WA.

Andrews, L.C., 1992, \An analytical model for the refractive index power

spectrum and its application to optical scintillations in the atmosphere," J.

Mod. Opt., 39:1849-1853.

Beland, R.R., 1993, \Propagation through Atmospheric Optical Turbulence,"

chapter 2 of volume 2 of The Infrared & Electro-Optical Systems Handbook, Atmospheric Propagation of Radiation, F.G. Smith, ed., SPIE Optical

Engineering Press, Bellingham, WA, pp. 157-232.

Blackadar, A.K., 1962, \Vertical Distribution of Wind and Turbulent Exchange

in a Neutral Atmosphere," J. Geophys. Res., 67:3095.

Churnside, J.H., 1990, \A spectrum of refractive-index turbulence in the

turbulent atmosphere," J. Mod. Opt., 37:13-16.

Cliord, S.F., 1978, \The Classical Theory of Wave Propagation in a Turbulent

Medium," chapter 2 of Laser Beam Propagation in the Atmosphere, J.W.

Strohbehn, ed., Springer-Verlag, Berlin.

Davis, C.A., and D.L. Walters, 1994, \Atmospheric inner-scale eects on

normalized irradiance variance," Appl. Opt., 33:8406-8411.

Ellerbroek, B.L., 1997, \Including outer scale eects in zonal adaptive optics

calculations," Appl. Opt., 36:9456-9467.

Finardi, S., F. Trombetti, F. Tampieri, G. Brusasca, 1995, \An Assessment of

Mixing-Length Closure Schemes for Models of Turbulent Boundary Layers over

Complex Terrain," Boundary Layer Meteorology, 73:343-356.

Frehlich, R., 1992, \Laser scintillation measurements of the temperature

spectrum in the atmospheric surface layer," J. Atmos. Sci., 49:1494-1509.

Frehlich, R., 2000, \Simulation of laser propagation in a turbulent atmosphere,"

Appl. Opt., 39:393-397.

Frenzen, P., and C.A. Vogel, 1995, \On the Magnitude and Apparent Range

of Variation of the von Karman Constant in the Atmospheric Surface Layer,"

Boundary Layer Meteorol., 72:371-392.

39
Fried, D.L., 1966, \Optical Resolution Through a Randomly Inhomogeneous

Medium for Very Long and Very Short Exposures," J Opt Soc Am, 56:1372-

1379.

Gaskill, J.D., 1978, Linear Systems, Fourier Transforms, and Optics, J. Wiley

& Sons, Inc., New York, NY.

Goodman, J.W., 1985, Statistical Optics, J. Wiley & Sons, Inc., New York, NY.

Gradshteyn, I.S., and I.M. Ryzhik, 1980, Table of Integrals, Series, and Products, translated and edited by Alan Jerey, Academic Press, Inc., Orlando, FL.

Hill, R.J., 1978, \Models of the scalar spectrum for turbulent advection," J.

Fluid Mech., 88:541-562.

Hill, R.J., and S.F. Cliord, 1978, \Modied Spectrum of Atmospheric

Turbulence Fluctuations and Its Application to Optical Propagation," J Opt

Soc Am, 68:892-899.

Hill, R.J., S.F. Cliord, and R.S. Lawrence, 1980, \Refractive Index and

Absorption Fluctuations in the Infrared Caused by Temperature, Humidity, and

Pressure Fluctuations," J. Opt. Soc. Am., 70:1182-1205.

Hill, R.J., and S.F. Cliord, 1981, \Theory of saturation of optical scintillation

by strong turbulence for arbitrary refractive-index spectra," J. Opt. Soc. Am., 71:675-686.

Hill, R.J., 1997, \Algorithms for obtaining atmospheric surface-layer 
uxes from

scintillation measurements", J. Atmos. Oceanic Technol., 14:456-467.

Hinze, J.O., 1987, Turbulence, McGraw-Hill Inc., New York, NY.

Ishimaru, A., 1978, Wave Propagation and Scattering in Random Media, Volume

2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing, Academic Press, New York, NY.

Kaimal, J.C., J.C. Wyngaard, Y. Izumi, and O.R. Cote, 1972, \Spectral

characteristics of surface-layer turbulence," Quarterly Journal of the Royal

Meteorological Society, 98:563-589.

Kaimal, J.C., and J.J. Finnegan, 1994, Atmospheric Boundary Layer Flows,

Their Structure and Measurement, Oxford University Press, New York, NY.

Kopeika, N.S., 1998, A System Engineering Approach to Imaging, SPIE Optical

Engineering Press, Bellingham, WA.

Kolmogorov, A.N., 1962, \A renement of previous hypotheses concerning the

local structure of turbulence in a viscous incompressible 
uid at high Reynolds

number," J. Fluid Mech., 13:82-85.

Kreyszig, E., 1972, Advanced Engineering Mathematics, J. Wiley and Sons, Inc.,

New York, NY.

40
Kunkel, K.E., and D.L. Walters, 1983, \Modeling the Diurnal Dependence of

the Optical Refractive Index Structure Parameter," J. Geoph. Res., 88:10,999-

11,004.

Lewellen, W.S., 1977, \Use of invariant modeling," chapter 9 of Handbook of

Turbulence, W. Frost and T.H. Moulden, eds., Plenum, New York, NY.

Martin, J.M., and S.M. Flatte, 1988, \Intensity images and statistics from

numerical simulation of wave propagation in 3D random media," Appl. Opt., 27:2111-2126.

Nelson, D.H., D.L. Walters, E.P. MacKerrow, M.J. Schmitt, C.R. Quick,

W.M. Porch, and R.R. Petrin, 2000, \Wave optics simulation of atmospheric

turbulence and re
ective speckle eects in CO2 lidar," Appl. Opt., 39:1857-1871.

Oke, T.R., 1978, Boundary Layer Climates, Methuen & Co. Ltd., London.

Panofsky, H.A., and J.A. Dutton, 1984, Atmospheric Turbulence: models and

methods for engineering applications, J. Wiley & Sons, New York, NY.

Rachele, H., and A. Tunick, 1994, \Energy Balance Model for Imagery and

Electromagnetic Propagation," J. Appl. Meteorology, 33:964-976.

Roddier, F., 1981, \The Eects of Atmospheric Turbulence in Optical

Astronomy," in Progress in Optics Vol. XIX, E.Wolf, ed., North-Holland.

Saman, P.G., 1967, \Large-Scale Structure of Homogeneous Turbulence," J.

Fluid Mech., 27:581.

Tatarskii, V.I., 1971, The Eects of the Turbulent Atmosphere on Wave

Propagation, translated for NOAA by Israel Program for Scientic Translations,

Jerusalem.

Taylor, G.I., 1938, \The spectrum of turbulence," Proc. Roy. Soc. London, A164:476-490.

Tofsted, D.H., 1991, Baseline Resolution of Atmosphere Related FADEWS

Modeling Issues, Volume 6, Scintil lation Distribution, Part 1 - Theoretical

Development, ASL-TR-0304-1, US Army Atmospheric Sciences Laboratory, White Sands Missile Range, NM 88002-5501.

Tofsted, D.H., and H.J. Auvermann, 1991, Baseline Resolution of Atmosphere

Related FADEWS Modeling Issues, Volume 9, Additional Model Improvements, ASL-TR-0302, US Army Atmospheric Sciences Laboratory, White Sands Missile

Range, NM 88002-5501.

Tofsted, D.H., 1992, \Outer-scale eects on beam wander and angle-of-arrival

variances," Appl. Opt., 31:5865-5870.

von Karman, T., 1948, \Progress in the statistical theory of turbulence," Proc.

Natl. Acad. Sci. U.S., 34:530-539.

Wesely, M.L., 1976, \The Combined Eect of Temperature and Humidity

Fluctuations on Refractive Index," J. Appl. Met., bf 15:43-49.

41
Yan, H., S. Li, D. Zhang, and S. Chan, 2000, \Numerical simulation of an

adaptive optics system with laser propagation in the atmosphere," Appl. Opt., 39:3023-3031.

Young, C.Y., L.C. Andrews, and A. Ishimaru, 1998, \Time-of-arrival


uctuations of a space-time Gaussian pulse in weak optical turbulence: an

analytic solution," Appl. Opt., 37:7655-7660.

Zhao, M., 1994, \A Planetary Boundary Layer Numerical Model Containing

Third-Order Derivative Terms," Boundary Layer Meteorology, 69:71-82.

42
Appendix

Integration Methods

43

In section 2.3.3 of this report, several integrations were performed. In this

appendix the methods used in performing these integrations are described. The

rst of these operations is related to the m family of functions introduced in

section 2.3.2. In 2.3.3, these functions must be integrated into one-dimensional

(1-D) form. Figure 1 contains a family of such integrated curves. Second, we

integrate these functions to generate the variance statistic 2n. These integrations can all be performed using the 3.259.3 result from Gradshteyn

and Ryzhik [1980] (hereafter, G&R):

Z1

0

x1(1 + xp)(1 + xp) dx =

1

p

=p B





p

;  +  



p 

2F1 ;



p

;  + ; 1 





;

(A  1)

where j arg j < , j arg j < , p > 0, and 0 < < < 2<( + ). The function

B(x; y) is the beta function,

B(x; y) = (x) (y)

(x + y)

: (A  2)

The 2F1 function is Gauss' hypergeometric function, given as,

2F1(; ; 
; z)=1+ 


  1

z +

 ( + 1)  ( + 1)


 (
 + 1)  1  2

z

2 + ::: (A  3)

In performing the evaluation of 2n begin by considering the integral,

2n =

Z1

0

n() 4 2 d: (A  4)

Removing constant terms from the integral, we have two resulting integrations

to perform:

Z1

0

( La)2 L

11=3 a

[1 + ( La)2]17=6 2 d = L2=3 a Z1

0

u4

[1 + u2]17=6 du; (A  5)

where u =  La, and,

Z1

0

( Lb)4 L

11=3

b

[1 + ( Lb)2]23=6 2 d = L

2=3

b Z1

0

u6

[1 + u2]23=6 du; (A  6)

45
where u =  Lb. We can write these two integrals in the more general form,

I (n) = Z1

0

x2n

(1 + x2)n+5=6 dx (A  7)

This integral can be solved using the cited G&R formula where the following

substitutions are introduced:  =  = 1, p = 2,  = 2n + 1,  = ,  +  =

n + 5=6. All the components here are real, and the nal condition is met where

2n + 1 < 2n + 5=3. Hence, we obtain the result,

I (n) = 1

2



n + 1

2  



1

3 



n + 5

6  : (A  8)

Here, the hypergeometric function's main argument (1  =) evaluates to zero,

and hence the function itself evaluates to unity and drops out of the nal

expression.

To evaluate the transformation from three-dimensional (3-D) spectra to 1-D, we

begin by considering the integral:

2 Z1

0

n q

2

1 + 2

r

2 r dr : (A  9)

Again, removing constants, we nd the following two integrals at the heart of

the problem:

Z1

0

2

1 + 2r  L2a L11=3 a

[1 + 2

1 L2a + 2

r L2a]17=6 r dr = L5=3 a Z1

0

v2 + u2

[1 + v2 + u2]17=6 u du; (A  10)

where u = rLa and v = 1La; and,

Z1

0

2

1 + 2r 2 L2

b L

11=3

b

[1 + 2

1 L2

b + 2

r L2

b

]23=6 r dr = L5=3

b Z1

0

v2 + u22

[1 + v2 + u2]23=6 u du; (A  11)

where u = rLb and v = 1Lb. To simplify these integrals we can remove the

v

2 dependence from the denominators:

Z1

0

v2 + u2

[1 + v2 + u2]17=6 u du =

v2

[1 + v2]17=6 Z1

0

u

[1 + A u2]17=6 du

+

1

[1 + v2]17=6 Z1

0

u3

[1 + A u2]17=6 du;

(A  12)

46
where A = (1 + v

2

)1; and,

Z1

0

v2 + u22

[1 + v2 + u2]23=6 u du =

v

4

[1 + v2]23=6 Z1

0

u

[1 + A u2]23=6 du

+

2v2

[1 + v2]23=6 Z1

0

u3

[1 + A u2]23=6 du

+

1

[1 + v2]23=6 Z1

0

u5

[1 + A u2]23=6 du;

(A  13)

where A = (1 + v2)1. These forms are thus seen to devolve to the solution of the following general

integral:

Z1

0

x2m+1

(1 + A x2)n+5=6 dx; (A  14)

where 0  m  n  1. This form can also be solved using the previously

mentioned G&R result. Here, we use  = 2m + 2, p = 2,  =  = A,  = , and  +  = n + 5=6. In general, we nd a result,

Z1

0

x2m+1

(1 + A x2)n+5=6 dx =

1

2

(1 + v

2

)

(m+1) (m + 1) (n  m  1=6)

(n + 5=6) : (A  15)

47

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US ARMY DUGWAY PROVING GRD 1 

STEDP3 

DUGWAY UT 84022-5000 
53 

USTRADOC 1 

ATCD FA 

FT MONROE VA 23651-5170 

WSMR TECH LIBRARY BR 1 

STEWS IM IT 

WSMR NM 88002 

US ARMY RESEARCH LAB 1 

AMSRL D DR D SMITH 

2800 POWDER MILL RD 

ADELPHI MD 20783-1197 

US ARMY CECOM 1 

INFORMATION & INTELLIGENCE 

WARFARE DIRECTORATE 

ATTN AMSEL RD IW IP 

FORT MONMOUTH NJ 07703-5211 

ARMY RESEARCH LABORATORY 21 

AMSRL CI E 

ATTN D TOFSTED 

WSMR NM 88002-5501 

Record copy 1 

TOTAL 77