Abstract

We introduce a model for the power spectrum of stratospheric turbulence that is based on data from in situ measurements and on results of the theory of saturated internal gravity waves. We then study the effect of that stratospheric turbulence on the scintillation and the coherence of starlight and on the degradation of star image. It is shown that, because the stratospheric phase structure function is approximately quadratic over a distance comparable with the aperture of a large telescope, stratospheric turbulence does not significantly degrade short-exposure images of star but does degrade long-exposure ones. The influence of stratosphere on star image degradation is more important in the IR than in the visible range because of the specific dependence of the stratospheric coherence radius on the light wavelength. Star image motion and blurring are determined by different characteristics of the atmosphere and cannot be explained solely on the basis of the Kolmogorov model. The modified astronomical imaging theory predicts greater improvement in resolution as the result of tilt removal than does a conventional theory.

© 1995 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. A. M. Obukhov, “Structure of the temperature field in a turbulent flux,” Izv. Akad. Nauk SSSR Ser. Geogr. Geophiz. 13, 58–69 (1949).
  3. S. Corrsin, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence,” J. Appl. Phys. 22, 469–473 (1951).
    [CrossRef]
  4. A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.
  5. F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
    [CrossRef]
  6. E. M. Dewan, “Simulated modelling of internal gravity wave spectra,” Geophys. Res. Lett. 18, 1473–1476 (1991).
    [CrossRef]
  7. Ch. A. Hostetler, Ch. S. Gardner, “Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific,” J. Geophys. Res. 99, 1283–1302 (1994).
    [CrossRef]
  8. E. P. Salathe, R. B. Smith, “In situ observation of temperature microstructure above and below the tropopause,” J. Atmos. Sci. 49, 2032–2036 (1992).
    [CrossRef]
  9. A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).
  10. F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
    [CrossRef]
  11. A. S. Gurvich, A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromag. Waves Appl. 7, 1343–1353 (1993).
    [CrossRef]
  12. A. S. Gurvich, “Model of three-dimensional spectrum of local axisymmetric temperature inhomogeneities in stable stratified atmosphere,” Atmos. Ocean. Phys. 31, 344–349 (1995).
  13. S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
    [CrossRef]
  14. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Wave Propagation through Random Media, Vol. 4 of Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).
  15. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  16. A. S. Gurvich, “On the scattering of sound and radio waves by turbulent structures in the stratosphere,” Atmos. Ocean. Phys. 30, 3–12 (1994).
  17. M. J. Beran, A. M. Whitman, “Scintillation index calculations using an altitude-dependent structure constant,” Appl. Opt. 27, 2178–2182 (1988).
    [CrossRef] [PubMed]
  18. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  19. F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
    [CrossRef]
  20. T. S. McKechnie, “Atmospheric turbulence and the resolution limits of large ground-based telescopes: reply to comment,” J. Opt. Soc. Am. A 10, 2415–2417 (1993).
    [CrossRef]
  21. D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometic seeing measurements of Mt. Wilson: power spectra and outer scales,” Appl. Opt. 34, 1081–1096 (1995).
    [CrossRef] [PubMed]
  22. V. I. Tatarskii, V. U. Zavorotny, “Atmospheric turbulence and the resolution limits of large ground-based telescopes: comment,” J. Opt. Soc. Am. A 10, 2410–2414 (1993).
    [CrossRef]

1995 (2)

A. S. Gurvich, “Model of three-dimensional spectrum of local axisymmetric temperature inhomogeneities in stable stratified atmosphere,” Atmos. Ocean. Phys. 31, 344–349 (1995).

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometic seeing measurements of Mt. Wilson: power spectra and outer scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

1994 (3)

A. S. Gurvich, “On the scattering of sound and radio waves by turbulent structures in the stratosphere,” Atmos. Ocean. Phys. 30, 3–12 (1994).

Ch. A. Hostetler, Ch. S. Gardner, “Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific,” J. Geophys. Res. 99, 1283–1302 (1994).
[CrossRef]

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

1993 (3)

1992 (1)

E. P. Salathe, R. B. Smith, “In situ observation of temperature microstructure above and below the tropopause,” J. Atmos. Sci. 49, 2032–2036 (1992).
[CrossRef]

1991 (1)

E. M. Dewan, “Simulated modelling of internal gravity wave spectra,” Geophys. Res. Lett. 18, 1473–1476 (1991).
[CrossRef]

1990 (1)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

1989 (1)

F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
[CrossRef]

1988 (1)

1987 (1)

S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
[CrossRef]

1966 (1)

1965 (1)

1951 (1)

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

1949 (1)

A. M. Obukhov, “Structure of the temperature field in a turbulent flux,” Izv. Akad. Nauk SSSR Ser. Geogr. Geophiz. 13, 58–69 (1949).

Aleksandrov, A. P.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Armstrong, J. T.

Beran, M. J.

Buscher, D. F.

Colavita, M. M.

Corrsin, S.

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

Crochet, M.

F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
[CrossRef]

Dalaudier, F.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
[CrossRef]

Denison, C. S.

Dewan, E. M.

E. M. Dewan, “Simulated modelling of internal gravity wave spectra,” Geophys. Res. Lett. 18, 1473–1476 (1991).
[CrossRef]

Fried, D. L.

Fritts, D. S.

S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
[CrossRef]

Gardner, Ch. S.

Ch. A. Hostetler, Ch. S. Gardner, “Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific,” J. Geophys. Res. 99, 1283–1302 (1994).
[CrossRef]

Grechko, G. M.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Gurvich, A. S.

A. S. Gurvich, “Model of three-dimensional spectrum of local axisymmetric temperature inhomogeneities in stable stratified atmosphere,” Atmos. Ocean. Phys. 31, 344–349 (1995).

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

A. S. Gurvich, “On the scattering of sound and radio waves by turbulent structures in the stratosphere,” Atmos. Ocean. Phys. 30, 3–12 (1994).

A. S. Gurvich, A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromag. Waves Appl. 7, 1343–1353 (1993).
[CrossRef]

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Hostetler, Ch. A.

Ch. A. Hostetler, Ch. S. Gardner, “Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific,” J. Geophys. Res. 99, 1283–1302 (1994).
[CrossRef]

Hummel, C. A.

Johnston, K. J.

Kan, V.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Kon, A. I.

A. S. Gurvich, A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromag. Waves Appl. 7, 1343–1353 (1993).
[CrossRef]

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Wave Propagation through Random Media, Vol. 4 of Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

Manarov, M. Kh.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

McKechnie, T. S.

Monin, A. S.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

Mozurkewich, D.

Obukhov, A. M.

A. M. Obukhov, “Structure of the temperature field in a turbulent flux,” Izv. Akad. Nauk SSSR Ser. Geogr. Geophiz. 13, 58–69 (1949).

Pakhomov, A. I.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Quirrenbach, A.

Roddier, F.

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

Romanenko, Yu. V.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Wave Propagation through Random Media, Vol. 4 of Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

Salathe, E. P.

E. P. Salathe, R. B. Smith, “In situ observation of temperature microstructure above and below the tropopause,” J. Atmos. Sci. 49, 2032–2036 (1992).
[CrossRef]

Savchenko, S. A.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Serova, S. I.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

Shao, M.

Sidi, C.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
[CrossRef]

Smith, R. B.

E. P. Salathe, R. B. Smith, “In situ observation of temperature microstructure above and below the tropopause,” J. Atmos. Sci. 49, 2032–2036 (1992).
[CrossRef]

Smith, S. A.

S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, V. U. Zavorotny, “Atmospheric turbulence and the resolution limits of large ground-based telescopes: comment,” J. Opt. Soc. Am. A 10, 2410–2414 (1993).
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Wave Propagation through Random Media, Vol. 4 of Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

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

Titov, V. G.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

VanZandt, T. E.

S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
[CrossRef]

Whitman, A. M.

Yaglom, A. M.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

Zavorotny, V. U.

Adv. Space Res. (1)

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situ techniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

Appl. Opt. (2)

Atmos. Ocean. Phys. (2)

A. S. Gurvich, “On the scattering of sound and radio waves by turbulent structures in the stratosphere,” Atmos. Ocean. Phys. 30, 3–12 (1994).

A. S. Gurvich, “Model of three-dimensional spectrum of local axisymmetric temperature inhomogeneities in stable stratified atmosphere,” Atmos. Ocean. Phys. 31, 344–349 (1995).

Geophys. Res. Lett. (1)

E. M. Dewan, “Simulated modelling of internal gravity wave spectra,” Geophys. Res. Lett. 18, 1473–1476 (1991).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Geogr. Geophiz. (1)

A. M. Obukhov, “Structure of the temperature field in a turbulent flux,” Izv. Akad. Nauk SSSR Ser. Geogr. Geophiz. 13, 58–69 (1949).

Izv. Atmos. Ocean Phys. (1)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. Kh. Manarov, A. I. Pakhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, V. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean Phys. 26, 1–7 (1990).

J. Appl. Phys. (1)

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

J. Atmos. Sci. (2)

E. P. Salathe, R. B. Smith, “In situ observation of temperature microstructure above and below the tropopause,” J. Atmos. Sci. 49, 2032–2036 (1992).
[CrossRef]

S. A. Smith, D. S. Fritts, T. E. VanZandt, “Evidence for a saturated spectrum of atmospheric gravity waves,” J. Atmos. Sci. 44, 1404–1410 (1987).
[CrossRef]

J. Electromag. Waves Appl. (1)

A. S. Gurvich, A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromag. Waves Appl. 7, 1343–1353 (1993).
[CrossRef]

J. Geophys. Res. (1)

Ch. A. Hostetler, Ch. S. Gardner, “Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific,” J. Geophys. Res. 99, 1283–1302 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Radio. Sci. (1)

F. Dalaudier, M. Crochet, C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling result,” Radio. Sci. 24, 311–324 (1989).
[CrossRef]

Other (4)

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

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Wave Propagation through Random Media, Vol. 4 of Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989).

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Figures (2)

Fig. 1
Fig. 1

Comparison of the stratospheric and tropospheric coherence radii, ρ0s and ρ0t, for different wavelengths. Curves 1, 4, and 5 correspond to the tropospheric coherence scale. The IITT is equal to 2 × 10−13 m1/3 for curve 1, 10−12 m1/3 for curve 4, and 10−11 m1/3 for curve 5. Curves 2 and 3 correspond to the stratospheric coherence scale for L0s = 100 and 200 m, respectively.

Fig. 2
Fig. 2

Dependence of the ratio ρ0e0t, which characterizes the influence of stratospheric turbulence on the coherence of starlight, on the wavelength. The parameter IITT is equal to 10−11 m1/3 for curve 1, 10−12 m1/3 for curve 2, and 2 × 10−13 m1/3 for curve 3.

Tables (1)

Tables Icon

Table 1 Dependencies of the Tropospheric, Stratospheric, and Effective Coherence Radii on Wavelength

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
Φ T ( κ ) = Φ T ( 1 ) ( κ ) + Φ T ( 2 ) ( κ ) .
Φ T ( 2 ) ( κ ) = S T 2 κ 5 f G ( κ / κ 0 s ) ,
F I ( κ ) = F I ( 1 ) ( κ ) + F I ( 2 ) ( κ ) ,
F I ( 1 , 2 ) ( κ , L ) = 2 π k 2 0 L sin 2 ( κ 2 z / 2 k ) Φ n ( 1 , 2 ) ( κ , z ) d z ,
σ I 2 = 2 π 0 F I ( κ ) κ d κ ,
D s ( 1 , 2 ) ( ρ , L ) = 4 π 0 [ 1 J 0 ( κ ρ ) ] F 8 ( 1 , 2 ) ( κ , L ) κ d κ
F s ( 1 , 2 ) ( κ , L ) = 2 π k 2 0 L cos 2 ( κ 2 z / 2 k ) Φ n ( 1 , 2 ) ( κ , Z ) d z ,
D s ( 1 ) ( ρ ) = 2.91 k 2 0 C n 2 ( z ) d z ρ 5 / 3 ,
f G ( κ / κ 0 s ) = [ κ 2 / ( κ 0 s 2 + κ 2 ) ] 5 / 2 .
D s ( 2 ) ( ρ ) = ( 4 / 3 ) π 2 k 2 S n 0 2 H 0 κ 0 s 3 × [ 1 ( 1 + κ 0 s ρ ) exp ( κ 0 s ρ ) ] .
D s ( 2 ) ( ρ ) = ( 2 / 3 ) π 2 k 2 S n 0 2 H 0 κ 0 s 1 ρ 2 [ 1 + O ( κ 0 s ρ ) ] .
ρ 0 s = ( π 2 k 2 S n 0 2 H 0 κ 0 s 1 / 3 ) 1 / 2
γ ( ρ ) = exp { 1 / 2 [ D s ( 1 ) ( ρ ) + D s ( 2 ) ( ρ ) ] } .
D s ( 1 ) ( ρ 0 e ) + D s ( 2 ) ( ρ 0 e ) = 2 .
σ I 2 = 9.45 π 2 k 1 / 2 S n 0 2 H tr 5 / 2 Ψ ( 1 , 7 / 2 , 2 H tr / H 0 ) ,
ρ 0 e = ρ 0 t [ 1 + ρ 0 e 1 / 3 ( ρ 0 t / ρ 0 s ) 5 / 3 ρ 0 s 1 / 3 ] 3 / 5 ,
ρ 0 e / ρ 0 s = [ 1 + ( ρ 0 t / ρ 0 s ) 2 ] 1 / 2 f ( ρ 0 t / ρ 0 s ) ,

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