Abstract

The performance of large optical systems can be strongly influenced by the behavior of large-scale atmospheric turbulence. Previous optical phase difference data diverged from theoretical predictions based on von Kármán’s model. To establish a better model microthermal data over large spatial scales were collected at two sites with few terrain inhomogeneities, in unstable midday conditions. Spatial structure function and spectra are developed. A model spatial spectrum is fit to the data with parameters to set two power law dependencies and the transition rate between them. By working with spatial spectral data the assumption of frozen flow is avoided. The optimum spatial spectrum is Kolmogorov-like in the inertial range. Temporal spectral data compare favorably with the new model, which better describes the transition of turbulence scaling from small scales to large.

© 2008 Optical Society of America

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  2. D. P. Greenwood and D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, Air Force Rome Air Development Center In-House Technical ReportRADC-TR-74-19 (Rome Air Development Center, 1974). Available as a scanned document through NTIS and Storming Media: http://handle.dtic.mil/100.2/AD776294.
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    [CrossRef]
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    [CrossRef]
  11. J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
    [CrossRef]
  12. M. Nakanishi, “Improvement of the Mellor-Yamada turbulence closure model based on large-eddy simulation data,” Boundary-Layer Meteorol. 99, 349-378 (2001).
    [CrossRef]
  13. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers,” C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941). Reprinted in Ref. .
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  15. A. M. Obukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 59-69 (1949).
  16. S. Panchev, Random Functions and Turbulence (Pergamon, 1971).
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    [CrossRef]
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    [CrossRef]
  21. J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
    [CrossRef]
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    [CrossRef]
  23. G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London A164, 476-490 (1938).
  24. H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
    [CrossRef]
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  30. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541-562 (1978).
    [CrossRef]
  31. 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]
  32. J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
    [CrossRef]

2007

A. D. Wheelon, N. Short, and C. H. Townes, “Low-frequency behavior of turbulence fluctuations at Mount Wilson Observatory,” Astrophys. J., Suppl. Ser. 172, 720-731 (2007).
[CrossRef]

2001

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

M. Nakanishi, “Improvement of the Mellor-Yamada turbulence closure model based on large-eddy simulation data,” Boundary-Layer Meteorol. 99, 349-378 (2001).
[CrossRef]

1995

1978

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541-562 (1978).
[CrossRef]

1977

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]

1976

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

C. S. Gardner, “Effects of random path fluctuations on the accuracy of laser ranging systems,” Appl. Opt. 15, 2539-2545 (1976).
[CrossRef] [PubMed]

1972

G. W. Reinhardt and S. A. Collins, Jr., “Outer-scale effects in turbulence-degraded light-beam spectra,” J. Opt. Soc. Am. 62, 1526-1530 (1972).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

A. J. Huber and R. P. Urtz, “Experimental determination of phase structure function at 10.6μm,” J. Opt. Soc. Am. 62, 1340A (1972).

1971

1968

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

1960

G. S. Golitsin, “On the structure of turbulence in the small-scale range,” Appl. Math. Mech. 24, 1705-1713 (1960).
[CrossRef]

1958

H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
[CrossRef]

1949

T. von Kármán and C. C. Lin, “On the concept of similarity in the theory of isotropic turbulence,” Rev. Mod. Phys. 21, 516-519 (1949). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 179-186.
[CrossRef]

A. M. Obukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 59-69 (1949).

1948

T. von Kármán, “Progress in the statistical theory of turbulence,” J. Mar. Res. 7, 252-264 (1948). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 161-174.

1941

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers,” C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941). Reprinted in Ref. .

1938

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London A164, 476-490 (1938).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of U.S. Department of Commerce Applied Mathematic Series (U.S. Department of Commerce, 1968).

Armstrong, J. T.

Buscher, D. F.

Caughey, S. J.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

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.

Colavita, M. M.

Collins, S. A.

Coté, O. R.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

Cramer, H. E.

H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
[CrossRef]

Denison, C. S.

Friedlander, S. K.

S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 151-161.

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]

Gardner, C. S.

Golitsin, G. S.

G. S. Golitsin, “On the structure of turbulence in the small-scale range,” Appl. Math. Mech. 24, 1705-1713 (1960).
[CrossRef]

Greenwood, D. P.

D. P. Greenwood and D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, Air Force Rome Air Development Center In-House Technical ReportRADC-TR-74-19 (Rome Air Development Center, 1974). Available as a scanned document through NTIS and Storming Media: http://handle.dtic.mil/100.2/AD776294.

Haugen, D. A.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

Hill, R.

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541-562 (1978).
[CrossRef]

Huber, A. J.

A. J. Huber and R. P. Urtz, “Experimental determination of phase structure function at 10.6μm,” J. Opt. Soc. Am. 62, 1340A (1972).

Hummel, C. A.

Izumi, Y.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

Johnston, K. J.

Kaimal, J. C.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

Kimmel, S. J.

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers,” C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941). Reprinted in Ref. .

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]

Lin, C. C.

T. von Kármán and C. C. Lin, “On the concept of similarity in the theory of isotropic turbulence,” Rev. Mod. Phys. 21, 516-519 (1949). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 179-186.
[CrossRef]

T. von Kármán and C. C. Lin, “On the statistical theory of isotropic turbulence,” in Advances in Applied Mechanics, Vol. 2, R.von Mises and T.von Kármán, eds. (Academic, 1951), pp. 1-19.
[CrossRef]

Mozurkewich, D.

Nakanishi, M.

M. Nakanishi, “Improvement of the Mellor-Yamada turbulence closure model based on large-eddy simulation data,” Boundary-Layer Meteorol. 99, 349-378 (2001).
[CrossRef]

Obukhov, A. M.

A. M. Obukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 59-69 (1949).

Ochs, G. R.

G. R. Ochs, A Resistance Thermometer for Measurement of Rapid Air Temperature Fluctuations, ESSA Tech. Rep. IER47-ITSA 46 (U.S. Department of Commerce, 1967).

G. R. Ochs, National Oceanic and Atmospheric Administration, Boulder, Colo. (personal communication, 1973).

Otte, M.

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

Panchev, S.

S. Panchev, Random Functions and Turbulence (Pergamon, 1971).

Panofsky, H. A.

H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
[CrossRef]

Quirrenbach, A.

Rao, V. R. K.

H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
[CrossRef]

Readings, C. J.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

Reinhardt, G. W.

Seaman, N.

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

Shao, M.

Short, N.

A. D. Wheelon, N. Short, and C. H. Townes, “Low-frequency behavior of turbulence fluctuations at Mount Wilson Observatory,” Astrophys. J., Suppl. Ser. 172, 720-731 (2007).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of U.S. Department of Commerce Applied Mathematic Series (U.S. Department of Commerce, 1968).

Strohbehn, J. W.

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Tarazano, D. O.

D. P. Greenwood and D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, Air Force Rome Air Development Center In-House Technical ReportRADC-TR-74-19 (Rome Air Development Center, 1974). Available as a scanned document through NTIS and Storming Media: http://handle.dtic.mil/100.2/AD776294.

Tatarskii, V. I.

V. I. Tatarskii, “The Effects of the Turbulent Atmosphere on Wave Propagation” (Israel Program for Scientific Translations, 1971), available from U.S. Department of Commerce, TT68-50464.

Taylor, G. I.

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London A164, 476-490 (1938).

Topper, L.

S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 151-161.

Townes, C. H.

A. D. Wheelon, N. Short, and C. H. Townes, “Low-frequency behavior of turbulence fluctuations at Mount Wilson Observatory,” Astrophys. J., Suppl. Ser. 172, 720-731 (2007).
[CrossRef]

Urtz, R. P.

A. J. Huber and R. P. Urtz, “Experimental determination of phase structure function at 10.6μm,” J. Opt. Soc. Am. 62, 1340A (1972).

Voitsekhovich, V. V.

von Kármán, T.

T. von Kármán and C. C. Lin, “On the concept of similarity in the theory of isotropic turbulence,” Rev. Mod. Phys. 21, 516-519 (1949). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 179-186.
[CrossRef]

T. von Kármán, “Progress in the statistical theory of turbulence,” J. Mar. Res. 7, 252-264 (1948). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 161-174.

T. von Kármán and C. C. Lin, “On the statistical theory of isotropic turbulence,” in Advances in Applied Mechanics, Vol. 2, R.von Mises and T.von Kármán, eds. (Academic, 1951), pp. 1-19.
[CrossRef]

Wheelon, A. D.

A. D. Wheelon, N. Short, and C. H. Townes, “Low-frequency behavior of turbulence fluctuations at Mount Wilson Observatory,” Astrophys. J., Suppl. Ser. 172, 720-731 (2007).
[CrossRef]

A. D. Wheelon, Electromagnetic Scintillation, I. Geometrical Optics (Cambridge U. Press, 2001).

Wyngaard, J. C.

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

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. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

J. C. Wyngaard, “On surface layer turbulence,” in Workshop on Micrometeorology, D.A.Haugen, ed. (American Meteorological Society, 1973), Chap. 3.

Appl. Math. Mech.

G. S. Golitsin, “On the structure of turbulence in the small-scale range,” Appl. Math. Mech. 24, 1705-1713 (1960).
[CrossRef]

Appl. Opt.

Astrophys. J., Suppl. Ser.

A. D. Wheelon, N. Short, and C. H. Townes, “Low-frequency behavior of turbulence fluctuations at Mount Wilson Observatory,” Astrophys. J., Suppl. Ser. 172, 720-731 (2007).
[CrossRef]

Boundary-Layer Meteorol.

M. Nakanishi, “Improvement of the Mellor-Yamada turbulence closure model based on large-eddy simulation data,” Boundary-Layer Meteorol. 99, 349-378 (2001).
[CrossRef]

C. R. (Dokl.) Acad. Sci. URSS

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds' numbers,” C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941). Reprinted in Ref. .

Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz.

A. M. Obukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 59-69 (1949).

J. Atmos. Sci.

J. C. Kaimal, J. C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey, and C. J. Readings, “Turbulence structure in the convective boundary layer,” J. Atmos. Sci. 33, 2152-2168 (1976).
[CrossRef]

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.

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541-562 (1978).
[CrossRef]

J. Mar. Res.

T. von Kármán, “Progress in the statistical theory of turbulence,” J. Mar. Res. 7, 252-264 (1948). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 161-174.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

J. W. Strohbehn, “Line-of-sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Proc. R. Soc. London

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London A164, 476-490 (1938).

Q. J. R. Meteorol. Soc.

H. A. Panofsky, H. E. Cramer, and V. R. K. Rao, “The relation between Eulerian time and space spectra,” Q. J. R. Meteorol. Soc. 84, 270-273 (1958).
[CrossRef]

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,” Q. J. R. Meteorol. Soc. 98, 563-589 (1972).
[CrossRef]

Radio Sci.

J. C. Wyngaard, N. Seaman, S. J. Kimmel, and M. Otte, “Concepts, observations and simulation of refractive index turbulence in the lower atmosphere,” Radio Sci. 36, 643-669 (2001).
[CrossRef]

Rev. Mod. Phys.

T. von Kármán and C. C. Lin, “On the concept of similarity in the theory of isotropic turbulence,” Rev. Mod. Phys. 21, 516-519 (1949). Reprinted in S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 179-186.
[CrossRef]

Other

J. C. Wyngaard, “On surface layer turbulence,” in Workshop on Micrometeorology, D.A.Haugen, ed. (American Meteorological Society, 1973), Chap. 3.

A. D. Wheelon, Electromagnetic Scintillation, I. Geometrical Optics (Cambridge U. Press, 2001).

S. K. Friedlander and L. Topper, Turbulence (Interscience, 1961), pp. 151-161.

T. von Kármán and C. C. Lin, “On the statistical theory of isotropic turbulence,” in Advances in Applied Mechanics, Vol. 2, R.von Mises and T.von Kármán, eds. (Academic, 1951), pp. 1-19.
[CrossRef]

S. Panchev, Random Functions and Turbulence (Pergamon, 1971).

V. I. Tatarskii, “The Effects of the Turbulent Atmosphere on Wave Propagation” (Israel Program for Scientific Translations, 1971), available from U.S. Department of Commerce, TT68-50464.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of U.S. Department of Commerce Applied Mathematic Series (U.S. Department of Commerce, 1968).

A.Erdelyi, ed., Tables of Integral Transforms II (McGraw Hill, 1954).

D. P. Greenwood and D. O. Tarazano, A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, Air Force Rome Air Development Center In-House Technical ReportRADC-TR-74-19 (Rome Air Development Center, 1974). Available as a scanned document through NTIS and Storming Media: http://handle.dtic.mil/100.2/AD776294.

G. R. Ochs, A Resistance Thermometer for Measurement of Rapid Air Temperature Fluctuations, ESSA Tech. Rep. IER47-ITSA 46 (U.S. Department of Commerce, 1967).

G. R. Ochs, National Oceanic and Atmospheric Administration, Boulder, Colo. (personal communication, 1973).

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Figures (14)

Fig. 1
Fig. 1

Normalized structure function data from mission 210873 and best theoretical fit using von Kármán’s spectrum. Note the divergence of data from theory at scales greater than 1/5 the outer scale.

Fig. 2
Fig. 2

Normalized temperature spectral data and theoretical curve based on von Kármán’s spectrum. (Abscissa is in units of logarithm of frequency in hertz. This is the case for all temporal spectra shown here.) Individual spectra were normalized prior to averaging for this final data plot. Divergence of data from theory in low frequencies is attributable to the spectral model, the frozen-flow assumption, or both.

Fig. 3
Fig. 3

Normalized temperature difference spectral data and theoretical curve based on von Kármán’s spectrum. (Again, abscissa is in units of logarithm of frequency in hertz.) Note that divergence in low frequencies is likely attributable to the frozen-flow assumption.

Fig. 4
Fig. 4

Modeled temperature spatial spectrum for q = 2 , 0 < s p < 2 . Similar curves can be derived for other values of q: smaller q’s give a smoother transition, larger q’s a more abrupt knee (not shown). For any value of q (here q = 2 ), an increase in s p results in a rounder transition at the knee of the curve.

Fig. 5
Fig. 5

Normalized temperature structure function for the model when q is set to 2. Not shown, but as with the spectra, smaller q’s give a rounder knee; larger q’s a sharper knee. Similarly, small s p values give sharper knees; larger ones give smoother transitions.

Fig. 6
Fig. 6

Normalized temperature structure function data (Griffiss AFB mission 210873) and best-fit theoretical curve based on a proposed model with q = 2 , p = 11 6 , and s = 1 . Clearly there is a better fit than seen in Fig. 1, at least with this data set, with a more gradual transition from the inertial subrange to large scales. Conditions for these data collection are given in Table 1.

Fig. 7
Fig. 7

Normalized temperature temporal spectra (single T) for the model with q = 2 . Earlier observations regarding the knee of the curve and the parameter q apply here as well, with smaller values of s p giving a more abrupt knee.

Fig. 8
Fig. 8

Single probe temperature spectral data (mission 210873), and the theoretical model curve based on using q = 2 , p = 11 6 , and s = 1 (as with Fig. 6 where the structure function was plotted). This demonstrates excellent fit.

Fig. 9
Fig. 9

Normalized temperature difference spectral data (mission 210873), and the theoretical model curve for q = 2 , p = 11 6 , s = 1 , and θ = 0 . Agreement may seem to break down in the lowest frequencies, but since the difference spectrum is largely model-independent in these frequencies, we attribute this to the frozen-flow assumption breaking down in conditions of high wind speed and direction variations.

Fig. 10
Fig. 10

Single probe temperature spectral data (mission 200373, Kirtland AFB), 33 m altitude, and a theoretical curve based on the proposed model ( q = 2 , p = 11 6 , s = 1 ) . This demonstrates excellent data agreement with the model.

Fig. 11
Fig. 11

Normalized temperature structure function data and theoretical model curve (mission 040973, Griffiss AFB). Excellent agreement is found with the parameters chosen earlier: q = 2 , p = 11 6 , s = 1 .

Fig. 12
Fig. 12

Normalized temperature structure function data and model curve for q = 2 , p = 11 6 , and s = 1 (mission 011073, Griffiss AFB), showing the best agreement of all the data shown.

Fig. 13
Fig. 13

Single-probe temperature spectral data and model for q = 2 , p = 11 6 , and s = 1 (mission 011073, Griffiss AFB).

Fig. 14
Fig. 14

Temperature-difference spectral data and model for q = 2 , p = 11 6 , and s = 1 . Note excellent agreement even in the lowest frequencies, well below f 1 , which is attributable to the best applicability of the model in low wind speed and direction variations.

Tables (3)

Tables Icon

Table 1 Conditions Recorded for Griffiss AFB Mission 210873

Tables Icon

Table 2 Conditions Recorded for Kirtland AFB Mission 200373

Tables Icon

Table 3 Conditions Recorded for Griffiss AFB Missions 040973 and 011073

Equations (48)

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Φ T ( κ ) κ 11 3 .
Φ T ( κ ) = 0.0330 C T 2 [ κ 2 + ( 1.071 Λ 0 ) 2 ] 11 6 ,
D T ( r ) = C T 2 r 2 3 .
Φ T ( κ ) κ 11 3 exp [ ( κ κ M ) 2 ] ,
Φ T ( κ ) = 0.0330 C T 2 ( κ 2 + L 0 2 ) 11 6 exp [ ( κ κ M ) 2 ] .
D T ( r ) = ( T ( r 1 ) T ( r 2 ) ) 2 ,
D T ( r ) = 8 π 0 [ 1 sinc ( κ r ) ] Φ T ( κ ) κ 2 d κ .
D T ( r ) = C T 2 r 2 3 [ 1.0468 ( r L 0 ) 2 3 0.62029 ( r L 0 ) 1 3 K 1 3 ( r L 0 ) ] ,
D T ( r Λ 0 ) = C T 2 r 2 3 ,
D T ( Λ 0 r ) = C T 2 Λ 0 2 3 .
R ( τ ) = 0 F ( f ) cos ( 2 π f τ ) d f .
F ( f ) = 4 π W ( 2 π f ) .
F T ( f ) = 8 π 2 ν 1 2 π f ν Φ T ( κ ) κ d κ .
F T ( f ) = 0.07308 C T 2 ν 2 3 [ f 2 + f 1 2 ] 5 6 ,
F Δ T ( f ; r ) = 2 d τ cos ( 2 π f τ ) R Δ τ ( τ , r ) ,
R Δ τ ( τ ; r ) = D T ( ν τ ) + 1 2 D T ( r + ν τ ) + 1 2 D T ( r ν τ ) .
R Δ T ( τ ; r ) = D T ( ν τ ) + 1 2 D T ( ( r + ν τ cos θ ) 2 + ( ν τ sin θ ) 2 ) + 1 2 D T ( ( r ν τ cos θ ) 2 + ( ν τ sin θ ) 2 ) ,
F Δ T ( f ; r ) = 8 π 0 d κ Φ T ( κ ) κ 2 d τ cos ( 2 π f τ ) [ 2 sinc ( κ ν τ ) sinc ( κ ( r + ν τ cos θ ) 2 + ( ν τ sin θ ) 2 ) sinc ( κ ( r ν τ cos θ ) 2 + ( ν τ sin θ ) 2 ) ] .
F Δ T ( f ; r ) = 16 π 2 ν 1 2 π f ν d κ Φ T ( κ ) κ [ 1 cos ( 2 π f r ν 1 cos θ ) J 0 ( r sin θ κ 2 ( 2 π f ν ) 2 ) ] .
F Δ T ( f ; r ) = 40 π 9 Γ ( 1 3 ) C T 2 ν 1 ( 2 π f ν ) 5 3 ,
0 d u u [ u 2 + 1 + ( f 1 f ) 2 ] 11 6 [ 1 cos ( f cos θ f 2 ) J 0 ( f u sin θ f 2 ) ] ,
F Δ T ( f ) = 3.1272 C T 2 r 5 3 ν 1 β θ = π 2 5 3 [ 1 cos ( f cos θ f 2 ) [ 1 G ( β ) ] ] ,
1 G ( β ) = 5 3 ( β 2 ) 5 6 1 Γ ( 11 6 ) K 5 6 ( β ) ,
β 2 = [ 1 + ( f 1 f ) 2 ] ( f sin θ f 2 ) 2 ,
F Δ T ( f f 1 , f 2 ) = 5.829 C T 2 r 5 3 ν ,
F Δ T ( f 1 , f 2 f ) = 0.14616 ν 2 3 C T 2 f 5 3 .
f Δ T ( f 1 , f 2 f ) = 2 F T ( f 1 f ) .
F Δ T ( f f 1 , f 2 ) = 3.1272 C T 2 L 0 5 3 ν 1 ( f f 2 ) 2 2 ,
F Δ T ( f 1 f f 2 ) = 3.1272 C T 2 r 5 3 ν 1 ( f f 2 ) 1 3 2 ,
F Δ T ( f 1 f ) = 3.1272 C T 2 r 5 3 ν 1 2 ( f f 2 ) 5 3 sin 2 ( f ( 2 f 2 ) ) .
Φ T ( κ ) = c [ ( κ L 0 ) q + ( κ L 0 ) s ] p ,
D T ( ) = 8 π 0 Φ T ( κ ) κ 2 d κ < .
D T ( ) = 8 π c L 0 3 ( q s ) Γ ( p ) Γ ( 3 s p q s ) Γ ( q p 3 q s ) .
Φ T ( κ ) = c [ κ L 0 + 1 ] q p .
D T ( r L 0 ) = c r q p 3 L 0 q p 8 π 2 csc [ π ( 2 q p ) ] Γ ( q p 1 ) sin [ π ( q p 2 ) 2 ] .
D T ( r L 0 ) = C T * 2 r p q 3 .
c = [ C T * 2 L 0 p q ( 8 π 2 ) ] Γ ( p q 1 ) sin [ π ( 2 p q ) ] csc [ π ( p q 2 ) 2 ] .
( Λ 0 L 0 ) p q 3 = Γ ( 3 s p q s ) Γ ( q p 3 q s ) Γ ( p q 1 ) ( q s ) π Γ ( p ) sin [ π ( 2 p q ) ] csc [ π ( p q 2 ) 2 ] .
ν L 0 2 8 π 2 c F T ( f ) 1 ( q s ) Γ ( p ) Γ ( 2 s p q s ) Γ ( q p 2 q s ) 1 2 s p ( f f 1 ) 2 s p ,
ν L 0 2 8 π 2 c F T ( f ) 1 ( p q 2 ) ( f f 1 ) 2 p q .
Φ T ( κ ) = 0.0030 C T 2 ( κ 2 + κ L 0 ) 11 6 .
D T ( r ) = C T 2 r 2 3 [ 1.1078 ( r L 0 ) 2 3 4.6173 ( r L 0 ) 5 3 Im U ( 1 6 , 2 3 , i r L 0 ) ] .
D T ( r L 0 ) = C T 2 r 2 3 , D T ( L 0 r ) = 1.1078 L 0 2 3 C T 2 ,
D T ( r L 0 ) D T ( L 0 r ) = 0.9027 ( r L 0 ) 2 3 .
F T ( f ) = 0.07307 C T 2 ν 2 3 f 5 3 F 1 2 ( 11 6 , 5 3 ; 8 3 ; f 1 f ) ,
lim f 0 F T ( f ) = 13.919 C T 2 ν 1 L 0 5 3 [ 1 1.1232 ( f 1 f ) 1 6 ] ,
lim f F T ( f ) = 0.07307 C T 2 ν 2 3 f 5 3 .
lim f 0 F Δ T ( f ) = 5.8276 ν 1 C T 2 ( r sin θ ) 5 3 [ 1 1.045 ( r L 0 1 sin θ ) 1 3 ] .

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