Abstract

The power spectrum of the difference angle of arrival of two parallel rays through turbulent air is computed and the results compared to data and to earlier theoretical work on the subject. The present theory predicts an f−2/3 power law in the inertial subrange, and thus differs significantly from Tatarski’s, which predicts an f−2/3 power law. Nevertheless, it fits the existing data in the f−2/3 region common to both theories. The interferometer situation introduces a filter factor that is unimportant for most vertical interferometers in turbulent air, except at very low frequencies.

© 1973 Optical Society of America

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References

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  1. M. A. Kallistratova and A. I. KonIzv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100 (1966).
  2. V. I. Tatarski, Effects of the Turbulent Atmosphere on Wave Propagation (translated for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971), available from U.S. Dept. of Commerce NTIS, Springfield, Va. 22151.
  3. The term “inertial subrange” can also be used in describing portions of the power spectrum. Whereas this term describes the wave-number range L0−1< K< l0−1in the refractive-index spatial spectrum, it describes the frequency range UL0−1< ω< Ul0−1in the power spectrum; U is the physically significant velocity determining the temporal characteristics, e.g., under Taylor’s hypothesis, and l0′, L0are the scales of turbulence.
  4. A. J. Favre, J. Appl. Mech. 32, 241 (1965).
    [Crossref]
  5. S. F. Clifford, J. Opt. Soc. Am. 61, 1285 (1971).
    [Crossref]
  6. Handbook of Mathematical Functions, edited by M. Abramovitz and I. Stegun, Natl. Bur. Std. (U.S.) Appl. Math. Series 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).
  7. A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).
  8. I am indebted to D. L. Fried for a critical remark clarifying this point.

1971 (1)

1968 (1)

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).

1966 (1)

M. A. Kallistratova and A. I. KonIzv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100 (1966).

1965 (1)

A. J. Favre, J. Appl. Mech. 32, 241 (1965).
[Crossref]

Clifford, S. F.

Favre, A. J.

A. J. Favre, J. Appl. Mech. 32, 241 (1965).
[Crossref]

Gurvich, A. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).

Kallistratova, M. A.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).

M. A. Kallistratova and A. I. KonIzv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100 (1966).

Kon, A. I.

M. A. Kallistratova and A. I. KonIzv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100 (1966).

Tatarski, V. I.

V. I. Tatarski, Effects of the Turbulent Atmosphere on Wave Propagation (translated for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971), available from U.S. Dept. of Commerce NTIS, Springfield, Va. 22151.

Time, N. S.

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (2)

M. A. Kallistratova and A. I. KonIzv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100 (1966).

A. S. Gurvich, M. A. Kallistratova, and N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1360 (1968).

J. Appl. Mech. (1)

A. J. Favre, J. Appl. Mech. 32, 241 (1965).
[Crossref]

J. Opt. Soc. Am. (1)

Other (4)

Handbook of Mathematical Functions, edited by M. Abramovitz and I. Stegun, Natl. Bur. Std. (U.S.) Appl. Math. Series 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

V. I. Tatarski, Effects of the Turbulent Atmosphere on Wave Propagation (translated for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971), available from U.S. Dept. of Commerce NTIS, Springfield, Va. 22151.

The term “inertial subrange” can also be used in describing portions of the power spectrum. Whereas this term describes the wave-number range L0−1< K< l0−1in the refractive-index spatial spectrum, it describes the frequency range UL0−1< ω< Ul0−1in the power spectrum; U is the physically significant velocity determining the temporal characteristics, e.g., under Taylor’s hypothesis, and l0′, L0are the scales of turbulence.

I am indebted to D. L. Fried for a critical remark clarifying this point.

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Figures (1)

Fig. 1
Fig. 1

Normalized angle-of-arrival power spectrum as a function of f/fT = 2πfL0/UT for two values of κmL0. The thick dashed curve represents an average of data from Ref. 7.

Equations (18)

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θ = k 1 T φ .
δθ = 1 2 0 L d z [ T δ ( ϱ z ) ] ρ = ρ / 2 ρ = + ρ / 2 = ( i / 4 π 2 ) 0 L d z d E ( K , z ) K sin ( K · ϱ / 2 ) ,
Φ ( K , τ ) = d 3 Δ r δ ( r , t ) δ * ( r + Δ r , t + τ ) e i K · Δ r ,
R ( τ ) = ( L / 8 π 2 ) d 2 K K 2 Φ ( K ; τ ) [ 1 cos ( K · ϱ ) ] .
Φ ( K ; τ ) = Φ ( K ) exp [ i K · U T τ 4 3 ( K Δ U τ ) 2 ] .
Φ ( K ) 32 π γ C n 2 ( K 2 + L 0 2 ) 11 / 6 exp ( K 2 / κ m 2 ) , γ 0.033 π 2 0.325 ,
W ( ω ) = 2 0 d τ R ( τ ) cos ω τ .
W ( ω ) = 8 γ C n 2 L L 0 1 3 [ W ( ω , 5 6 ) W ( ω , 11 / 6 ) ] , W ( ω , q ) = ω 1 0 d y cos y 0 d x ( 1 + x ) q × exp { x [ ( y Δ ω / 2 ω ) 2 + ( κ m L 0 ) 2 ] } × { J 0 ( x y ω T / ω ) 1 2 J 0 ( x y + ω T / ω ) 1 2 J 0 ( x y ω T / ω ) } , y ± 2 ( y ± ξ ω / ω T ) 2 + ( ξ ω / ω T ) 2 ,
ω T = U T / L 0 , Δ ω = 4 Δ U / L 0 3 Ω T = κ m U T , Δ Ω = 4 κ m Δ U / 3 .
W ( ω , q ) = ω 1 0 d y cos y 0 d x ( 1 + x ) q × exp { x [ ( y Δ ω / 2 ω ) 2 + ( κ m L 0 ) 2 ] } × { J 0 ( x y ω T / ω ) cos ( ξ ω / ω T ) × J 0 ( x [ y 2 + ( ξ ω / ω T ) 2 ] 1 2 ω T / ω ) } .
W ( ω , q ) = π 1 2 ω T 1 ( 1 + ω 2 / ω T 2 ) 1 2 q × e ω 2 / Ω T 2 U [ 1 2 , 3 2 q , ( ω 2 + ω T 2 ) / Ω T 2 ] × { 1 m = 0 1 m ! [ ξ 2 4 ( 1 + ω 2 ω T 2 ) ] m × U [ 1 2 + m , 3 2 q + m , ( ω 2 + ω T 2 ) / Ω T 2 ] U [ 1 2 , 3 2 q , ( ω 2 + ω T 2 ) / Ω T 2 ] × cos ( ξ ω / ω T ) } ,
W ( ω , q ) Γ ( q 1 2 ) Γ ( q ) · π ω T ( 1 + ω 2 ω T 2 ) 1 2 q for ω Ω T κ m L 0 π ω T ( 1 + ω 2 ω T ) q e ω 2 / Ω T 2 for ω Ω T ,
W ( ω , q ) = Γ ( q 1 2 ) Γ ( q ) π ω T ( 1 + ω 2 / ω T 2 ) 1 2 q × { 1 2 3 2 q Γ ( q 1 2 ) [ ξ 2 ( 1 + ω 2 ω T 2 ) ] q / 2 1 4 × K 1 2 q ( ξ [ 1 + ω 2 ω T 2 ] 1 2 ) cos ( ξ ω / ω T ) } ,
W ( ω , 5 6 ) = Γ ( 1 3 ) π Δ ω ( ω Δ ω ) 2 3 M ( 1 6 , 1 , ω T 2 / Δ ω 2 ) × { 1 2 cos ( ξ ω / ω T ) Γ ( 1 3 ) m = 0 1 m ! ( ω ξ 2 ω T ) m + 1 3 K m 1 3 ( ξ ω / ω T ) × [ M ( 1 6 + m , 1 , ω T 2 / Δ ω 2 ) / M ( 1 6 , 1 , ω T 2 / Δ ω 2 ) ] } ,
W ( ω = 5 6 ) Γ ( 1 3 ) Γ ( 5 6 ) π ω T ( ω ω T ) 2 3 { } for Δ ω ω T ω Γ ( 1 3 ) π Δ ω ( ω Δ ω ) 2 3 { } for ω T Δ ω ω ,
W ( ω ) 8 γ C n 2 L L 0 1 3 [ Γ ( 1 3 ) / Γ ( 5 6 ) Γ ( 4 3 ) / Γ ( 11 / 6 ) ] × π Δ ω I 0 ( ω T 2 / 2 Δ ω 2 ) exp ( ω T 2 / 2 Δ ω 2 ) { } ,
f W ( f ) / 0 d f W ( f ) = 2 f W ( f ) / R ( 0 )
R ( 0 ) = Γ ( 1 3 ) · ( κ m L 0 ) 1 3 · ( 8 γ C n 2 L L 0 1 3 ) .