Abstract

A semiempirical theory is developed which relates the refractive-index-structure parameter Cn2 in the lowest few tens of meters to the vertical gradients of wind speed and temperature and to a stability parameter. This provides an indirect method of determining Cn2, or its temperature counterpart CT2, and the method is successfully applied to recent data which include directly measured CT2 values. The height dependence of the structure parameter is also determined, and the predicted z−4/3 profile under unstable conditions agrees with recent data in the free atmosphere at heights up to 500 m.

© 1971 Optical Society of America

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References

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  1. V. L. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961), Ch. 10.
  2. H. A. Panofsky, J. Geophys. Res. 73, 6047 (1968).
    [Crossref]
  3. A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. 13, 58 (1949).
  4. J. O. Hinze, Turbulence (McGraw–Hill, New York, 1959), p. 189.
  5. The constant 0.25 stands for 2(2π)−1Γ(5/3) sin(π/3). Note that we use a range of 0 to ∞ for κ1, whereas a −∞ to +∞ range is used on p. 25 of Ref. 1.
  6. J. L. Lumley, Phys. Fluids 8, 1056 (1965).
    [Crossref]
  7. S. Corrsin, J. Appl. Phys. 22, 469 (1951).
    [Crossref]
  8. H. A. Panofsky, Radio Sci. 4, 1143 (1969).
    [Crossref]
  9. J. C. Wyngaard and O. R. Coté, J. Atmos. Sci. 28, 190 (1971).
    [Crossref]
  10. J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
    [Crossref]
  11. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, New York, 1964).
  12. J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
    [Crossref]
  13. L. R. Tsvang, Radio Sci. 4, 1175 (1969).
    [Crossref]
  14. E. W. Peterson, Quart. J. Roy. Meteorol. Soc. 95, 561 (1969)
    [Crossref]

1971 (3)

J. C. Wyngaard and O. R. Coté, J. Atmos. Sci. 28, 190 (1971).
[Crossref]

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
[Crossref]

1969 (3)

L. R. Tsvang, Radio Sci. 4, 1175 (1969).
[Crossref]

E. W. Peterson, Quart. J. Roy. Meteorol. Soc. 95, 561 (1969)
[Crossref]

H. A. Panofsky, Radio Sci. 4, 1143 (1969).
[Crossref]

1968 (1)

H. A. Panofsky, J. Geophys. Res. 73, 6047 (1968).
[Crossref]

1965 (1)

J. L. Lumley, Phys. Fluids 8, 1056 (1965).
[Crossref]

1951 (1)

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[Crossref]

1949 (1)

A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. 13, 58 (1949).

Bradley, E. F.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

Businger, J. A.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

Corrsin, S.

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[Crossref]

Coté, O. R.

J. C. Wyngaard and O. R. Coté, J. Atmos. Sci. 28, 190 (1971).
[Crossref]

J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
[Crossref]

Hinze, J. O.

J. O. Hinze, Turbulence (McGraw–Hill, New York, 1959), p. 189.

Izumi, Y.

J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
[Crossref]

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

Lumley, J. L.

J. L. Lumley, Phys. Fluids 8, 1056 (1965).
[Crossref]

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, New York, 1964).

Obukhov, A. M.

A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. 13, 58 (1949).

Panofsky, H. A.

H. A. Panofsky, Radio Sci. 4, 1143 (1969).
[Crossref]

H. A. Panofsky, J. Geophys. Res. 73, 6047 (1968).
[Crossref]

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, New York, 1964).

Peterson, E. W.

E. W. Peterson, Quart. J. Roy. Meteorol. Soc. 95, 561 (1969)
[Crossref]

Tatarski, V. L.

V. L. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961), Ch. 10.

Tsvang, L. R.

L. R. Tsvang, Radio Sci. 4, 1175 (1969).
[Crossref]

Wyngaard, J. C.

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
[Crossref]

J. C. Wyngaard and O. R. Coté, J. Atmos. Sci. 28, 190 (1971).
[Crossref]

Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. (1)

A. M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geograf. Geofiz. 13, 58 (1949).

J. Appl. Phys. (1)

S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[Crossref]

J. Atmos. Sci. (3)

J. C. Wyngaard and O. R. Coté, J. Atmos. Sci. 28, 190 (1971).
[Crossref]

J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, J. Atmos. Sci. 28, 181 (1971).
[Crossref]

J. C. Wyngaard, O. R. Coté, and Y. Izumi, J. Atmos. Sci. 28, 1171 (1971).
[Crossref]

J. Geophys. Res. (1)

H. A. Panofsky, J. Geophys. Res. 73, 6047 (1968).
[Crossref]

Phys. Fluids (1)

J. L. Lumley, Phys. Fluids 8, 1056 (1965).
[Crossref]

Quart. J. Roy. Meteorol. Soc. (1)

E. W. Peterson, Quart. J. Roy. Meteorol. Soc. 95, 561 (1969)
[Crossref]

Radio Sci. (2)

L. R. Tsvang, Radio Sci. 4, 1175 (1969).
[Crossref]

H. A. Panofsky, Radio Sci. 4, 1143 (1969).
[Crossref]

Other (4)

J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence (Wiley–Interscience, New York, 1964).

V. L. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961), Ch. 10.

J. O. Hinze, Turbulence (McGraw–Hill, New York, 1959), p. 189.

The constant 0.25 stands for 2(2π)−1Γ(5/3) sin(π/3). Note that we use a range of 0 to ∞ for κ1, whereas a −∞ to +∞ range is used on p. 25 of Ref. 1.

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Figures (3)

Fig. 1
Fig. 1

The dimensionless temperature-structure parameter vs Richardson number.

Fig. 2
Fig. 2

A comparison of directly measured CT2 values with values obtained with the indirect method.

Fig. 3
Fig. 3

The vertical profile of CT2.

Tables (1)

Tables Icon

Table I Values of f3(Ri) in the relation CT2 = z4/3(∂ Θ ¯/∂z)2f3(Ri).

Equations (21)

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C T 2 = ( 79 × 10 - 6 P / T 2 ) - 2 C n 2
[ T ( x ) - T ( x + r ) ] 2 av = C T 2 r 2 3 ,
Φ T ( κ 1 ) = 0.25 C T 2 κ 1 - 5 / 3 ,
0 Φ T ( κ 1 ) d κ 1 = ( T - T ¯ ) 2 av .
Φ T ( κ 1 ) = β 1 N - 1 3 κ 1 - 5 / 3 .
C T 2 = 3.2 N - 1 3 .
Ri = ( g / T ¯ ) ( Θ ¯ / d z ) ( Ū / z ) 2 ,
z / u * 3 = f 1 ( Ri ) , N z u * ( Θ ¯ / z ) 2 = f 2 ( Ri ) .
C T 2 z 4 3 ( Θ ¯ / z ) 2 = 3.2 f 2 f 1 1 3 = f 3 ( Ri ) .
C T 2 = z 4 3 ( Θ ¯ / z ) 2 f 3 ( Ri ) .
C T 2 = ( 4 ) 4 3 ( - 0.2 ) 2 ( 1.9 ) = 0.48 ° C 2 m - 2 3 .
Ū z | z 3 Δ Ū Δ z = Ū ( z 2 ) - Ū ( z 1 ) z 3 ln ( z 2 / z 1 ) ,
L = - u * 3 T ¯ / k g Q ,
z u * Q Θ ¯ z = z T * Θ ¯ z = g 1 ( z / L ) , Ri = g 2 ( z / L ) .
C T 2 = T * 2 z - 2 3 g 3 ( z / L ) .
g 3 = 4.9 [ 1 - 7 ( z / L ) ] - 2 3 0 z / L , g 3 = 4.9 [ 1 + 2.75 ( z / L ) ] 0 z / L .
g 3 4 3 ( - z / L ) - 2 3 ,
C T 2 4 3 T * 2 ( - L ) 2 3 z - 4 3 .
C T 2 4 3 k 2 3 ( T ¯ g ) 2 3 Q 4 3 z - 4 3 .
C T 2 z 4 3 g 2 3 / Q 4 3 T ¯ 2 3 = a .
C T 2 = a ( T ¯ / g ) 2 3 Q 4 3 z - 4 3 ,