Abstract

The outer scale of turbulence L0 has been calculated from values of the refractive-index structure coefficient CN2 obtained from spatio-angular correlation measurements of stellar scintillation. It is found that L0 ≤ 5 m for a large range of observations in France, U.S.A., and Chile and that its dependence on altitude Z follows the same general form at all these sites. The prediction of CN2(Z) profiles is shown to be feasible utilizing standard meteorological radiosonde data and this L0(Z) curve. A simple model based on dimensional analysis and a more complicated stochastic model are compared, but the latter appears to have no advantage.

© 1988 Optical Society of America

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References

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  1. B. R. Bean, E. J. Dutton, Radio Meteorology, Nat. Bur. Stand. U.S. Monogr.92 (1966) (Dover, New York, 1968).
  2. C. E. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
    [CrossRef]
  3. R. S. Lawrence, “Remote Sensing by Optical Line-of-Sight Propagation,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (WPL/ERL, Boulder, CO, 1972), Chap. 25.
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).
  5. V. I. Tatarski, “Effects of the Turbulent Atmosphere on Wave Propagation,” Israel Program of Scientific Translation, Jerusalem (1971).
  6. E. E. Gossard, “Refractive Index Variance and its Height Distribution in Different Air Masses,” Radio Sci. 12, 89 (1977).
    [CrossRef]
  7. J. Vernin, F. Roddier, “Experimental Determination of Two-Dimensional Spatio-Temporal Power Spectra of Stellar Light Scintillation,” J. Opt. Soc. Am. 63, 270 (1973).
    [CrossRef]
  8. J. Vernin, M. Azouit, “Traitement d’image adapté au Speckle Atmosphérique, II,” J. Opt. Paris 14, 131 (1983).
    [CrossRef]
  9. T. E. Vanzandt, D. S. Gage, J. M. Warnock, “An Improved Model for Calculation of Profiles of CN2 and ∊ in the Free Atmosphere from Background Profiles Wind, Temperature and Humidity,” in Twentieth Conference on Radar Meteorology (American Meteorological Society, Boston, 1981)
  10. F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
    [CrossRef]
  11. J. M. Mariotti, G. P. Di Benedetto, “Pathlength Stability of Synthetic Aperture Telescopes. The Case of the 25 cm CERGA Interferometer,” I.A.U. Colloq. 79, Garching Apl. 9–12 (1984).
  12. J. Barat, “On the Contamination of Stratospheric Turbulence Measurements by Wind Shear,” J. Atmos. Sci. 41, 819 (1984).
    [CrossRef]
  13. J. M. Warnock, T. E. Vanzandt, “A Statistical Model to Estimate the Refractivity Turbulence Structure Constant CN2 in the Free Atmosphere,” NOAA Tech. Memo. ERL. AL-10Aeronomy Lab., Boulder, CO (1985).
  14. C. E. Coulman, “Vertical Profiles of Small-Scale Temperature Structure in the Atmosphere,” Boundary-Layer Meteorol. 4, 169 (1973).
    [CrossRef]
  15. J. L. Bufton, “Correlation of Microthermal Turbulence Data with Meteorological Soundings in the Troposphere,” J. Atmos. Sci. 30, 83 (1973).
    [CrossRef]
  16. R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).
  17. G. C. Valley “Long and Short-Term Strehl Ratios for Turbulence with Finite Inner and Outer Scales,” Appl. Opt. 18, 984 (1979).
    [CrossRef] [PubMed]

1985

C. E. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

1984

J. Barat, “On the Contamination of Stratospheric Turbulence Measurements by Wind Shear,” J. Atmos. Sci. 41, 819 (1984).
[CrossRef]

1983

J. Vernin, M. Azouit, “Traitement d’image adapté au Speckle Atmosphérique, II,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

1981

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

1979

1977

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

E. E. Gossard, “Refractive Index Variance and its Height Distribution in Different Air Masses,” Radio Sci. 12, 89 (1977).
[CrossRef]

1973

J. Vernin, F. Roddier, “Experimental Determination of Two-Dimensional Spatio-Temporal Power Spectra of Stellar Light Scintillation,” J. Opt. Soc. Am. 63, 270 (1973).
[CrossRef]

C. E. Coulman, “Vertical Profiles of Small-Scale Temperature Structure in the Atmosphere,” Boundary-Layer Meteorol. 4, 169 (1973).
[CrossRef]

J. L. Bufton, “Correlation of Microthermal Turbulence Data with Meteorological Soundings in the Troposphere,” J. Atmos. Sci. 30, 83 (1973).
[CrossRef]

Azouit, M.

J. Vernin, M. Azouit, “Traitement d’image adapté au Speckle Atmosphérique, II,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

Barat, J.

J. Barat, “On the Contamination of Stratospheric Turbulence Measurements by Wind Shear,” J. Atmos. Sci. 41, 819 (1984).
[CrossRef]

Barletti, R.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Bean, B. R.

B. R. Bean, E. J. Dutton, Radio Meteorology, Nat. Bur. Stand. U.S. Monogr.92 (1966) (Dover, New York, 1968).

Bufton, J. L.

J. L. Bufton, “Correlation of Microthermal Turbulence Data with Meteorological Soundings in the Troposphere,” J. Atmos. Sci. 30, 83 (1973).
[CrossRef]

Ceppatelli, G.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Coulman, C. E.

C. E. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

C. E. Coulman, “Vertical Profiles of Small-Scale Temperature Structure in the Atmosphere,” Boundary-Layer Meteorol. 4, 169 (1973).
[CrossRef]

Di Benedetto, G. P.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength Stability of Synthetic Aperture Telescopes. The Case of the 25 cm CERGA Interferometer,” I.A.U. Colloq. 79, Garching Apl. 9–12 (1984).

Dutton, E. J.

B. R. Bean, E. J. Dutton, Radio Meteorology, Nat. Bur. Stand. U.S. Monogr.92 (1966) (Dover, New York, 1968).

Gage, D. S.

T. E. Vanzandt, D. S. Gage, J. M. Warnock, “An Improved Model for Calculation of Profiles of CN2 and ∊ in the Free Atmosphere from Background Profiles Wind, Temperature and Humidity,” in Twentieth Conference on Radar Meteorology (American Meteorological Society, Boston, 1981)

Gossard, E. E.

E. E. Gossard, “Refractive Index Variance and its Height Distribution in Different Air Masses,” Radio Sci. 12, 89 (1977).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).

Lawrence, R. S.

R. S. Lawrence, “Remote Sensing by Optical Line-of-Sight Propagation,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (WPL/ERL, Boulder, CO, 1972), Chap. 25.

Mariotti, J. M.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength Stability of Synthetic Aperture Telescopes. The Case of the 25 cm CERGA Interferometer,” I.A.U. Colloq. 79, Garching Apl. 9–12 (1984).

Paterno, L.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Righini, A.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Roddier, F.

Speroni, N.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Tatarski, V. I.

V. I. Tatarski, “Effects of the Turbulent Atmosphere on Wave Propagation,” Israel Program of Scientific Translation, Jerusalem (1971).

Valley, G. C.

Vanzandt, T. E.

J. M. Warnock, T. E. Vanzandt, “A Statistical Model to Estimate the Refractivity Turbulence Structure Constant CN2 in the Free Atmosphere,” NOAA Tech. Memo. ERL. AL-10Aeronomy Lab., Boulder, CO (1985).

T. E. Vanzandt, D. S. Gage, J. M. Warnock, “An Improved Model for Calculation of Profiles of CN2 and ∊ in the Free Atmosphere from Background Profiles Wind, Temperature and Humidity,” in Twentieth Conference on Radar Meteorology (American Meteorological Society, Boston, 1981)

Vernin, J.

Warnock, J. M.

T. E. Vanzandt, D. S. Gage, J. M. Warnock, “An Improved Model for Calculation of Profiles of CN2 and ∊ in the Free Atmosphere from Background Profiles Wind, Temperature and Humidity,” in Twentieth Conference on Radar Meteorology (American Meteorological Society, Boston, 1981)

J. M. Warnock, T. E. Vanzandt, “A Statistical Model to Estimate the Refractivity Turbulence Structure Constant CN2 in the Free Atmosphere,” NOAA Tech. Memo. ERL. AL-10Aeronomy Lab., Boulder, CO (1985).

Ann. Rev. Astron. Astrophys.

C. E. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. Astrophys. 23, 19 (1985).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

R. Barletti, G. Ceppatelli, L. Paterno, A. Righini, N. Speroni, “Astronomical Site Testing with Balloon-Borne Radiosondes,” Astron. Astrophys. 54, 649 (1977).

Boundary-Layer Meteorol.

C. E. Coulman, “Vertical Profiles of Small-Scale Temperature Structure in the Atmosphere,” Boundary-Layer Meteorol. 4, 169 (1973).
[CrossRef]

J. Atmos. Sci.

J. Barat, “On the Contamination of Stratospheric Turbulence Measurements by Wind Shear,” J. Atmos. Sci. 41, 819 (1984).
[CrossRef]

J. Atmos. Sci.

J. L. Bufton, “Correlation of Microthermal Turbulence Data with Meteorological Soundings in the Troposphere,” J. Atmos. Sci. 30, 83 (1973).
[CrossRef]

J. Opt. Paris

J. Vernin, M. Azouit, “Traitement d’image adapté au Speckle Atmosphérique, II,” J. Opt. Paris 14, 131 (1983).
[CrossRef]

J. Opt. Soc. Am.

Prog. Opt.

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

Radio Sci.

E. E. Gossard, “Refractive Index Variance and its Height Distribution in Different Air Masses,” Radio Sci. 12, 89 (1977).
[CrossRef]

Other

B. R. Bean, E. J. Dutton, Radio Meteorology, Nat. Bur. Stand. U.S. Monogr.92 (1966) (Dover, New York, 1968).

T. E. Vanzandt, D. S. Gage, J. M. Warnock, “An Improved Model for Calculation of Profiles of CN2 and ∊ in the Free Atmosphere from Background Profiles Wind, Temperature and Humidity,” in Twentieth Conference on Radar Meteorology (American Meteorological Society, Boston, 1981)

R. S. Lawrence, “Remote Sensing by Optical Line-of-Sight Propagation,” in Remote Sensing of the Troposphere, V. E. Derr, Ed. (WPL/ERL, Boulder, CO, 1972), Chap. 25.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978).

V. I. Tatarski, “Effects of the Turbulent Atmosphere on Wave Propagation,” Israel Program of Scientific Translation, Jerusalem (1971).

J. M. Mariotti, G. P. Di Benedetto, “Pathlength Stability of Synthetic Aperture Telescopes. The Case of the 25 cm CERGA Interferometer,” I.A.U. Colloq. 79, Garching Apl. 9–12 (1984).

J. M. Warnock, T. E. Vanzandt, “A Statistical Model to Estimate the Refractivity Turbulence Structure Constant CN2 in the Free Atmosphere,” NOAA Tech. Memo. ERL. AL-10Aeronomy Lab., Boulder, CO (1985).

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Figures (7)

Fig. 1
Fig. 1

(a) SCIDAR measurements of C N 2 made at Observatoire de Haute Provence, France plotted against M2 calculated by Eq. (7) from radiosonde temperature and pressure data obtained at Nimes, France for dates and times given in Table I. The points are so scattered that no attempt is made to fit a relationship to these data. Nevertheless, there seems to be a well-marked upper limit to the data which suggests that L0 does not exceed a few meters. The broken lines are marked with L0 in meters. (b) As for (a), but the data relate to SCIDAR observations at McDonald Observatory, TX and to radiosonde data from El Paso, TX for times and dates shown in Table I.

Fig. 2
Fig. 2

Some of the variance in Fig. 1 is attributable to the height dependence of L0. If L0 values are sorted into sets relating to 1-km thick slices of the atmosphere, a relationship given by Eq. (11) may be fitted. A systematic variation of L0 with altitude, therefore, seems to be confirmed. The histogram derived directly from the data is shown by broken lines.

Fig. 3
Fig. 3

SCIDAR measurements of C N 2 ( Z ) at Observatoire de Haute Provence are shown by the solid line. Radiosonde observations at Nimes (see Table I) have been used to calculate C N 2 ( Z ) from temperature and pressure data by means of model (A) (long dashes) and model (B) (short dashes). This example is typical of the degree of accord between models and measurements (Fig. 4) and relates to 10 Mar. 1985.

Fig. 4
Fig. 4

As for Fig. 3 but an example taken from measurements made at MacDonald Observatory (Table I). This represents the worst case of disagreement between models and observations in Europe and U.S.A. Observations made on 28 Oct. 1985.

Fig. 5
Fig. 5

As for Fig. 2 but relating to observations made at European Southern Observatory, La Silla, Chile (Table II). The curve has the same general form as that in Fig. 2 but a slightly higher peak value. This suggests that L0 not only has a small average value (i.e., 5 m or less) but also a fairly systematic dependence on altitude.

Fig. 6
Fig. 6

As for Fig. 3, but the data were obtained at ESO, La Silla, Chile (Table II), and the curve in Fig. 5 was used to represent L0(Z) in models (A) (long dashes) and (B) (short dashes). This is typical of the extent of agreement between C N 2 ( Z ) observed and C N 2 ( Z ) calculated by the models. Observations were made 14 and 15 Feb. 1986.

Fig. 7
Fig. 7

As for Fig. 6, but this example represents a case of serious disagreement between model calculations and observed results, especially in the 8–16-km altitude range. As in previous examples the stochastic model (B) seems to be prone to an overestimation of C N 2 by a factor of the order of 5. One possible hypothesis is that in the 8–16-km range the atmosphere was not truly turbulent, and if that were so the Kolmogorov theory would be inapplicable.

Tables (2)

Tables Icon

Table I Times, Dates, and Locations for SCIDAR Observations and Corresponding Radiosondes for Measurements Made in France (Upper Half); Similar Data for Observations in U.S.A. (Lower Half)

Tables Icon

Table II As for Table I, but the Data Refer to the European Southern Observatory, La Silla, Chile where Radiosonde Flights were Made from the Observatory Site

Equations (15)

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

D N ( r ¯ ) = [ N ( r ¯ 1 + r ¯ ) N ( r ¯ 1 ) ] 2 = C N 2 r 2 / 3
Δ N = N N ,
Δ N = δ N δ T Δ T + δ N δ e Δ e + δ N δ p Δ p ,
C N 2 = C T 2 [ 78 × 10 6 p T 2 ] 2 .
C N 2 = [ 78 × 10 6 p ( γ 1 ) T 2 ] 2 C T 2 ,
F N ( K ) = constant ( C N 2 K 5 / 3 ) ,
l 0 < 2 π / K < L 0 .
C N 2 = a M 2 L 0 4 / 3 ,
M = ( 78 × 10 6 p T ) δ ln θ δ Z .
θ = T ( 1000 / p ) 0 . 286
C N 2 = a M 2 F ( H 2 , S 2 ) .
R i = H 2 S 2 < 1 4 .
L 0 ( Z ) = 4 1 + ( Z 8500 2500 ) 2
L 0 ( Z ) = 5 1 + ( Z 7500 2000 ) 2
D S ( r ) = C S 2 r 2 / 3 , l 0 < r < L 0 ,

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