Abstract

Recent experiments reveal the high-wave-number form of the power spectrum of temperature fluctuations in turbulent flow. It is precisely this high-wave-number portion of the temperature spectrum that strongly affects optical propagation in the atmosphere. An accurate model of the spectra of advected quantities, such as temperature, has been developed and is applied here to optical propagation. An outstanding feature of the model and the observed temperature spectrum is a “bump” at high wave numbers. The accurate model of the temperature spectrum is used to compute the temperature structure function, the variance of log intensity as a function of Fresnel-zone size, the covariance function of log amplitude, the structure function of phase, as well as the phase coherence length. These results are compared with the predictions of Tatarskii’s spectrum. The bump in the temperature spectrum produces a corresponding bump in the temperature structure function, the variance of log intensity, and the structure function of phase. The accurate model is also used to determine the shape of the structure function of aerosol concentration fluctuations; it is found that this structure function varies as the logarithm of the separation distance for small separations.

© 1978 Optical Society of America

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References

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  1. F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
    [CrossRef]
  2. R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 547–567 (1977).
    [CrossRef]
  3. A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13, 58–69 (1949).
  4. S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
    [CrossRef]
  5. R. J. Hill, “Models of the scalar spectrum for turbulent advection” J. Fluid Mech. (to be published).
  6. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges” Radio Sci. (to be published).
  7. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65.
  8. G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid, Part I, General discussion and the case of small conductivity,” J. Fluid Mech. 5, 113–133 (1959).
    [CrossRef]
  9. H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
    [CrossRef]
  10. C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
    [CrossRef]
  11. C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11, 1612–1617 (1968).
    [CrossRef]
  12. R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11, 945–953 (1968).
    [CrossRef]
  13. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 46–48.
  14. M. J. Post and G. M. Lerfald, “Experimental measurements of atmospheric aerosol inhomogeneities,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.
  15. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  16. S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, and Margot H. Ackley, “Phase variations in atmospheric optical propagation,” J. Opt. Soc. Am. 61, 1279–1284 (1971).
    [CrossRef]
  17. S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.
  18. A. Consortini and L. Ronchi, “Some comments on the theory of E.M. propagation in a turbulent atmosphere,” Nuovo Cimento Lett. 2, 683–688 (1969).
    [CrossRef]
  19. A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
    [CrossRef]
  20. H. T. Yura, “Atmospheric turbulence induced laser beam spread,” Appl. Opt. 10, 2771–2772 (1971).
    [CrossRef] [PubMed]
  21. H. T. Yura, “Optical beam spread in a turbulent medium: effect of the outer scale of turbulence,” J. Opt. Soc. Am. 63, 107–109 (1973).
    [CrossRef]
  22. R.J. Hill, “Optical propagation in turbulent water” J. Opt. Soc. Am. (to be published).

1977 (2)

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 547–567 (1977).
[CrossRef]

1973 (2)

1971 (2)

1970 (2)

C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
[CrossRef]

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

1969 (1)

A. Consortini and L. Ronchi, “Some comments on the theory of E.M. propagation in a turbulent atmosphere,” Nuovo Cimento Lett. 2, 683–688 (1969).
[CrossRef]

1968 (3)

C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11, 1612–1617 (1968).
[CrossRef]

R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11, 945–953 (1968).
[CrossRef]

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

1959 (1)

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid, Part I, General discussion and the case of small conductivity,” J. Fluid Mech. 5, 113–133 (1959).
[CrossRef]

1951 (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[CrossRef]

1949 (1)

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13, 58–69 (1949).

Ackley, Margot H.

Batchelor, G. K.

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid, Part I, General discussion and the case of small conductivity,” J. Fluid Mech. 5, 113–133 (1959).
[CrossRef]

Bouricius, G. M. B.

Buffington, A.

S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Champagne, F. H.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Clifford, S. F.

Consortini, A.

A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
[CrossRef]

A. Consortini and L. Ronchi, “Some comments on the theory of E.M. propagation in a turbulent atmosphere,” Nuovo Cimento Lett. 2, 683–688 (1969).
[CrossRef]

Corrsin, S.

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[CrossRef]

Crawford, F. S.

S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Friehe, C. A.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Gibson, C. H.

C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
[CrossRef]

Grant, H. L.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

Hill, R. J.

R. J. Hill, “Models of the scalar spectrum for turbulent advection” J. Fluid Mech. (to be published).

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges” Radio Sci. (to be published).

Hill, R.J.

R.J. Hill, “Optical propagation in turbulent water” J. Opt. Soc. Am. (to be published).

Hirschsohn, I.

C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
[CrossRef]

Hughes, B. A.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

Kraichnan, R. H.

R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11, 945–953 (1968).
[CrossRef]

LaRue, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Leith, C. E.

C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11, 1612–1617 (1968).
[CrossRef]

Lerfald, G. M.

M. J. Post and G. M. Lerfald, “Experimental measurements of atmospheric aerosol inhomogeneities,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Lyon, R. R.

C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
[CrossRef]

Moilliet, A.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

Moroder, E.

Muller, R. A.

S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Oboukhov, A. M.

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13, 58–69 (1949).

Ochs, G. R.

Paulson, C. A.

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 547–567 (1977).
[CrossRef]

Pollaine, S.

S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Post, M. J.

M. J. Post and G. M. Lerfald, “Experimental measurements of atmospheric aerosol inhomogeneities,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

Ronchi, L.

A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
[CrossRef]

A. Consortini and L. Ronchi, “Some comments on the theory of E.M. propagation in a turbulent atmosphere,” Nuovo Cimento Lett. 2, 683–688 (1969).
[CrossRef]

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 46–48.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65.

Vogel, W. M.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

Williams, R. M.

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 547–567 (1977).
[CrossRef]

Wyngaard, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

Yura, H. T.

AIAA J. (1)

C. H. Gibson, R. R. Lyon, and I. Hirschsohn, “Reaction product fluctuations in a sphere wake,” AIAA J. 8, 1859–1863 (1970).
[CrossRef]

Appl. Opt. (1)

Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. (1)

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. i Geofiz. 13, 58–69 (1949).

J. Appl. Phys. (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
[CrossRef]

J. Atmos. Sci. (1)

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wyngaard, “Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34, 515–530 (1977).
[CrossRef]

J. Fluid Mech. (3)

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83, 547–567 (1977).
[CrossRef]

G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid, Part I, General discussion and the case of small conductivity,” J. Fluid Mech. 5, 113–133 (1959).
[CrossRef]

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–492 (1968).
[CrossRef]

J. Opt. Soc. Am. (3)

Nuovo Cimento Lett. (1)

A. Consortini and L. Ronchi, “Some comments on the theory of E.M. propagation in a turbulent atmosphere,” Nuovo Cimento Lett. 2, 683–688 (1969).
[CrossRef]

Phys. Fluids (2)

C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11, 1612–1617 (1968).
[CrossRef]

R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11, 945–953 (1968).
[CrossRef]

Proc. IEEE (1)

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Other (7)

S. Pollaine, A. Buffington, F. S. Crawford, and R. A. Muller, “A measurement of the phase structure function,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), pp. 46–48.

M. J. Post and G. M. Lerfald, “Experimental measurements of atmospheric aerosol inhomogeneities,” Technical Digest of the Topical Meeting on Optical Propagation Through Turbulence, Rain, and Fog; August 9–11, 1977, Boulder, Colorado. Available from: Optical Society of America, 2100 Pennsylvania Ave., N.W., Washington, D.C. 20037.

R. J. Hill, “Models of the scalar spectrum for turbulent advection” J. Fluid Mech. (to be published).

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges” Radio Sci. (to be published).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65.

R.J. Hill, “Optical propagation in turbulent water” J. Opt. Soc. Am. (to be published).

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Figures (10)

FIG. 1
FIG. 1

Spatial power spectrum ΦT of temperature fluctuations in air (ν/D = 0.72). The solid curve is the accurate model; the dashed curve is the Tatarskii model.

FIG. 2
FIG. 2

Temperature structure function from both the accurate model spectrum and the Tatarskii spectrum.

FIG. 3
FIG. 3

Same as Fig. 2 but with an outer scale spectral cutoff typical of a modest Reynolds number experiment.

FIG. 4
FIG. 4

Variance of log intensity for negligible saturation. Solid curve: spherical waves, accurate model; dashed curve: spherical waves, Tatarskii model; Δ: plane waves, accurate model; X: plane waves, Tatarskii model.

FIG. 5
FIG. 5

Covariance function of log amplitude for plane waves. Solid curves: accurate model; dashed curves: Tatarskii model.

FIG. 6
FIG. 6

Covariance function of log amplitude for plane waves. Solid curves: accurate model; dashed curves: Tatarskii model.

FIG. 7
FIG. 7

Structure function of phase for plane waves and no outer scale. Solid curves: accurate model; dashed curves: Tatarskii model.

FIG. 8
FIG. 8

Structure function of phase (plane waves, no outer scale) showing the ρ2 dependence at small ρ. Solid curves: accurate model; dashed curves: Tatarskii model.

FIG. 9
FIG. 9

Structure function of phase for plane waves and an outer scale cutoff. Solid curves: accurate model; dashed curves: Tatarskii model.

FIG. 10
FIG. 10

Phase coherence length scaled by both ρ1 and ρ2 for both an infinite and a finite outer scale. Solid curves: accurate model; dashed curves: Tatarskii model.

Equations (29)

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Φ T ( κ ) = ( 4 π ) - 1 β χ ɛ - 1 / 3 κ - 11 / 3 ,
Φ T ( κ ) = ( 4 π ) - 1 β χ ɛ - 1 / 3 κ - 11 / 3 exp ( - κ 2 / κ m 2 ) ,
d d κ H ( κ ) d d κ Φ T ( κ ) = 2 D κ 4 Φ T ( κ ) ,
H ( κ ) = ( 3 / 11 ) β - 1 ɛ 1 / 3 κ 14 / 3 [ ( κ / κ ) 2 b + 1 ] - 1 / ( 3 b ) ,
κ = 0.072 / η ,             b = 1.9.
C T 2 l 0 2 / 3 ( χ / 6 D ) l 0 2 .
D T ( r ) = 8 π 0 ( 1 - sin ( κ r ) κ r ) Φ T ( κ ) κ 2 d κ .
C T 2 = ( 9 / 10 ) Γ ( 1 / 3 ) β χ ɛ - 1 / 3 ,
l 0 / η = [ 27 β Γ ( 1 / 3 ) / 5 Pr ] 3 / 4 = 7.4             for air .
l 0 = [ 27 β Γ ( 1 / 3 ) / 5 ] 3 / 4 l C = 5.4 l C .
Φ s ( κ ) = ( 5 χ / 4 π Λ ) κ - 3 ( 1 + κ l s ) exp ( - κ l s ) ,
Λ ( 45 / 11 β ) ( κ η ) 2 / 3 ( ɛ / ν ) 1 / 2 = 1.0 ( ɛ / ν ) 1 / 2
l s ( 22 β / 3 ) 1 / 2 ( κ η ) - 1 / 3 l B = 5.5 l B .
D s ( r ) = 10 χ Λ 0 ( 1 - sin ( u r / l s ) u r / l s ) ( 1 + u ) u - 1 e - u d u = ( 5 χ / Λ ) ln [ 1 + ( r / l s ) 2 ] .
σ ln I 2 = 8 π 2 k 2 L 0 F ( x ) Φ n ( κ ) κ d κ ,
F ( x ) = 1 - x - 1 [ cos ( π x 2 / 2 ) C ( x ) + sin ( π x 2 / 2 ) S ( x ) ]
x = ( κ 2 L / 2 π k ) 1 / 2 ,
σ ln I 2 = A C n 2 k 7 / 6 L 11 / 6 ,
σ ˜ ln I 2 = σ ln I 2 / [ A C n 2 k 7 / 6 L 11 / 6 ] .
B χ ( ρ ) = 2 π 2 k 2 L 0 ( 1 - k κ 2 L sin κ 2 L k ) × J 0 ( κ ρ ) Φ n ( κ ) κ d κ .
D S ( ρ ) = 4 π 2 k 2 L 0 [ 1 - J 0 ( κ ρ ) ] × [ 1 + k κ 2 L sin ( κ 2 L k ) ] Φ n ( κ ) κ d κ .
D ˜ S ( ρ ) = D S ( ρ ) / 2.92 C n 2 k 2 L λ L 5 / 3 .
1 = 4 π 2 k 2 L 0 [ 1 - J 0 ( κ ρ 0 ) ] Φ n ( κ ) κ d κ .
( k 2 L C n 2 l 0 5 / 3 ) - 1 = 4 π 2 ( 0.033 / β ) ( l 0 / η ) - 11 / 3 × 0 [ 1 - J 0 ( κ ˜ ρ 0 / l 0 ) ] Φ ˜ T ( κ ˜ ) κ ˜ d κ ˜ .
Z i ( k 2 C n 2 l 0 5 / 3 ) - 1 .
Z i = 0.72 ( k 2 C n 2 l 0 5 / 3 ) - 1 ,
Z i = 0.67 ( k 2 C n 2 l 0 5 / 3 ) - 1 .
k 2 L C n 2 l 0 5 / 3 = 0.72 L / Z i ,
k 2 L C n 2 l 0 5 / 3 = 0.67 L / Z i .