Abstract

Experimental evidence and computational evidence suggest that the distribution of random segment lengths defined by the intersections of ray paths with the contorted and folded interface between two fluids mixed by turbulence follows a probability distribution with a Lévy law tail. Assuming that the two fluids have different optical properties, one finds that the statistics of light scattered by the mixing interface reflect the probability distribution for the random distances between the intersection points of straight ray paths with the interface. Examples of light-scattering calculations for limiting cases, including the planetary albedo problem and imaging through a transparent mixing layer, are discussed.

© 1996 Optical Society of America

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  1. K. R. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
    [Crossref]
  2. R. R. Prasad, K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (1990).
    [Crossref]
  3. G. F. Lane-Serff, “Investigation of the fractal structure of jets and plumes,” J. Fluid Mech. 249, 521–534 (1993).
    [Crossref]
  4. P. Flohr, D. Olivari, “Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet,” Physica D 76, 278–290 (1994).
    [Crossref]
  5. P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
    [Crossref] [PubMed]
  6. S. Lovejoy, “Area-perimeter relation for rain and cloud areas,” Science 216, 185–187 (1982).
    [Crossref] [PubMed]
  7. P. L. Miller, P. E. Dimotakis, “Stochastic geometric properties of scalar interfaces in turbulent jets,” Phys. Fluids A 3, 168–177 (1991).
    [Crossref]
  8. P. Lévy, Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, Paris, 1937).
  9. B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Wesley, Cambridge, 1954), p. 162.
  10. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  12. G. C. Pomraning, “A model for interface intensities in stochastic particle transport,” J. Quant. Spectrosc. Radiat. Transfer 46, 221–236 (1991).
    [Crossref]
  13. G. C. Pomraning, “Radiative transfer in random media with scattering,” J. Quant. Spectrosc. Radiat. Transfer 40, 479–487 (1988).
    [Crossref]
  14. G. C. Pomraning, “Classic transport problems in binary homogeneous markov statistical mixtures,” Transp. Theory Stat. 17, 595–613 (1988).
    [Crossref]
  15. C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
    [Crossref]
  16. D. Vanderhaegen, “Impact of a mixing structure on radiative transfer in random media,” J. Quant. Spectrosc. Radiat. Transfer 39, 333–337 (1988).
    [Crossref]
  17. D. Vanderhaegen, “Radiative transfer in statistically heterogeneous mixtures,” J. Quant. Spectrosc. Radiat. Transfer 36, 557–561 (1986).
    [Crossref]
  18. C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
    [Crossref]
  19. J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
    [Crossref]
  20. W. M. Irvine, “Diffuse reelection and transmission by clouds and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471–485 (1968).
    [Crossref]
  21. H. C. van de Hulst, Multiple Light Scattering Tables, Formulas and Application (Academic, New York, 1980).
  22. W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
    [Crossref]
  23. A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
    [Crossref] [PubMed]
  24. D. Mihalas, Stellar Atmospheres, 2nd ed. (Freeman, San Francisco, Calif., 1978).
  25. F. H. Shu, The Physics of Astrophysics, Vol. 1. Radiation (University Science, Mill Valley, Calif., 1991).
  26. W. B. Rossow, R. A. Schiffer, “ISCCP cloud data products,” Bull. Am. Meteorol. Soc. 72, 2–20 (1991).
    [Crossref]
  27. V. R. Taylor, L. L. Stowe, “Reflectance characteristics of uniform Earth and cloud surfaces derived from NIMBUS-7 ERB,” J. Geophys. Res. 89, 4987–4996 (1984).
    [Crossref]
  28. G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part II: group theory and simple closures,” J. Atmos. Sci. 45, 1837–1848 (1988).
    [Crossref]
  29. V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
    [Crossref] [PubMed]
  30. B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
    [Crossref]
  31. R. M. Cameron, M. Bader, R. E. Mobley, “Design and operation of the NASA 91.5 cm airborne telescope,” Appl. Opt. 10, 2011–2015 (1971).
    [Crossref] [PubMed]
  32. J. E. Craig, C. Allen, “Aero-optical turbulent boundary layer/shear layer experiment on the KC-135 aircraft revisited,” Opt. Eng. 24, 446–454 (1985).
  33. E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
    [Crossref]

1995 (1)

1994 (2)

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[Crossref]

P. Flohr, D. Olivari, “Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet,” Physica D 76, 278–290 (1994).
[Crossref]

1993 (2)

J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
[Crossref]

G. F. Lane-Serff, “Investigation of the fractal structure of jets and plumes,” J. Fluid Mech. 249, 521–534 (1993).
[Crossref]

1991 (4)

P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
[Crossref] [PubMed]

P. L. Miller, P. E. Dimotakis, “Stochastic geometric properties of scalar interfaces in turbulent jets,” Phys. Fluids A 3, 168–177 (1991).
[Crossref]

G. C. Pomraning, “A model for interface intensities in stochastic particle transport,” J. Quant. Spectrosc. Radiat. Transfer 46, 221–236 (1991).
[Crossref]

W. B. Rossow, R. A. Schiffer, “ISCCP cloud data products,” Bull. Am. Meteorol. Soc. 72, 2–20 (1991).
[Crossref]

1990 (2)

B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
[Crossref]

R. R. Prasad, K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (1990).
[Crossref]

1989 (1)

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

1988 (5)

G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part II: group theory and simple closures,” J. Atmos. Sci. 45, 1837–1848 (1988).
[Crossref]

G. C. Pomraning, “Radiative transfer in random media with scattering,” J. Quant. Spectrosc. Radiat. Transfer 40, 479–487 (1988).
[Crossref]

G. C. Pomraning, “Classic transport problems in binary homogeneous markov statistical mixtures,” Transp. Theory Stat. 17, 595–613 (1988).
[Crossref]

C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
[Crossref]

D. Vanderhaegen, “Impact of a mixing structure on radiative transfer in random media,” J. Quant. Spectrosc. Radiat. Transfer 39, 333–337 (1988).
[Crossref]

1986 (3)

D. Vanderhaegen, “Radiative transfer in statistically heterogeneous mixtures,” J. Quant. Spectrosc. Radiat. Transfer 36, 557–561 (1986).
[Crossref]

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

K. R. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[Crossref]

1985 (1)

J. E. Craig, C. Allen, “Aero-optical turbulent boundary layer/shear layer experiment on the KC-135 aircraft revisited,” Opt. Eng. 24, 446–454 (1985).

1984 (1)

V. R. Taylor, L. L. Stowe, “Reflectance characteristics of uniform Earth and cloud surfaces derived from NIMBUS-7 ERB,” J. Geophys. Res. 89, 4987–4996 (1984).
[Crossref]

1982 (1)

S. Lovejoy, “Area-perimeter relation for rain and cloud areas,” Science 216, 185–187 (1982).
[Crossref] [PubMed]

1980 (1)

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[Crossref]

1971 (1)

1968 (1)

W. M. Irvine, “Diffuse reelection and transmission by clouds and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471–485 (1968).
[Crossref]

Ahmad, E.

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Allen, C.

J. E. Craig, C. Allen, “Aero-optical turbulent boundary layer/shear layer experiment on the KC-135 aircraft revisited,” Opt. Eng. 24, 446–454 (1985).

Bader, M.

Barkstrom, B. R.

B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
[Crossref]

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Ben-David, A.

Cameron, R. M.

Cess, R. D.

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Constantin, P.

P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
[Crossref] [PubMed]

Craig, J. E.

J. E. Craig, C. Allen, “Aero-optical turbulent boundary layer/shear layer experiment on the KC-135 aircraft revisited,” Opt. Eng. 24, 446–454 (1985).

Dimotakis, P. E.

P. L. Miller, P. E. Dimotakis, “Stochastic geometric properties of scalar interfaces in turbulent jets,” Phys. Fluids A 3, 168–177 (1991).
[Crossref]

Flohr, P.

P. Flohr, D. Olivari, “Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet,” Physica D 76, 278–290 (1994).
[Crossref]

Gnedenko, B. V.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Wesley, Cambridge, 1954), p. 162.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Harrison, E. F.

B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
[Crossref]

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Hartmann, D.

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Irvine, W. M.

W. M. Irvine, “Diffuse reelection and transmission by clouds and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471–485 (1968).
[Crossref]

Klafter, J.

J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
[Crossref]

Kolmogorov, A. N.

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Wesley, Cambridge, 1954), p. 162.

Lane-Serff, G. F.

G. F. Lane-Serff, “Investigation of the fractal structure of jets and plumes,” J. Fluid Mech. 249, 521–534 (1993).
[Crossref]

Lee, R. B.

B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
[Crossref]

Levermore, C. D.

C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
[Crossref]

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

Lévy, P.

P. Lévy, Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, Paris, 1937).

Lovejoy, S.

S. Lovejoy, “Area-perimeter relation for rain and cloud areas,” Science 216, 185–187 (1982).
[Crossref] [PubMed]

Magee, E. P.

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[Crossref]

Meador, W. E.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[Crossref]

Meneveau, C.

K. R. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[Crossref]

Mihalas, D.

D. Mihalas, Stellar Atmospheres, 2nd ed. (Freeman, San Francisco, Calif., 1978).

Miller, P. L.

P. L. Miller, P. E. Dimotakis, “Stochastic geometric properties of scalar interfaces in turbulent jets,” Phys. Fluids A 3, 168–177 (1991).
[Crossref]

Minnis, P.

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Mobley, R. E.

Olivari, D.

P. Flohr, D. Olivari, “Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet,” Physica D 76, 278–290 (1994).
[Crossref]

Pomraning, G. C.

G. C. Pomraning, “A model for interface intensities in stochastic particle transport,” J. Quant. Spectrosc. Radiat. Transfer 46, 221–236 (1991).
[Crossref]

G. C. Pomraning, “Radiative transfer in random media with scattering,” J. Quant. Spectrosc. Radiat. Transfer 40, 479–487 (1988).
[Crossref]

G. C. Pomraning, “Classic transport problems in binary homogeneous markov statistical mixtures,” Transp. Theory Stat. 17, 595–613 (1988).
[Crossref]

C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
[Crossref]

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

Prasad, R. R.

R. R. Prasad, K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (1990).
[Crossref]

Procaccia, I.

P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
[Crossref] [PubMed]

Ramanathan, V.

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Rossow, W. B.

W. B. Rossow, R. A. Schiffer, “ISCCP cloud data products,” Bull. Am. Meteorol. Soc. 72, 2–20 (1991).
[Crossref]

Sanzo, D. L.

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

Schiffer, R. A.

W. B. Rossow, R. A. Schiffer, “ISCCP cloud data products,” Bull. Am. Meteorol. Soc. 72, 2–20 (1991).
[Crossref]

Shlesinger, M. F.

J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
[Crossref]

Shu, F. H.

F. H. Shu, The Physics of Astrophysics, Vol. 1. Radiation (University Science, Mill Valley, Calif., 1991).

Sreenivasan, K. R.

P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
[Crossref] [PubMed]

R. R. Prasad, K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (1990).
[Crossref]

K. R. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[Crossref]

Stephens, G. L.

G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part II: group theory and simple closures,” J. Atmos. Sci. 45, 1837–1848 (1988).
[Crossref]

Stowe, L. L.

V. R. Taylor, L. L. Stowe, “Reflectance characteristics of uniform Earth and cloud surfaces derived from NIMBUS-7 ERB,” J. Geophys. Res. 89, 4987–4996 (1984).
[Crossref]

Tatarski, V. I.

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

Taylor, V. R.

V. R. Taylor, L. L. Stowe, “Reflectance characteristics of uniform Earth and cloud surfaces derived from NIMBUS-7 ERB,” J. Geophys. Res. 89, 4987–4996 (1984).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas and Application (Academic, New York, 1980).

Vanderhaegen, D.

D. Vanderhaegen, “Impact of a mixing structure on radiative transfer in random media,” J. Quant. Spectrosc. Radiat. Transfer 39, 333–337 (1988).
[Crossref]

D. Vanderhaegen, “Radiative transfer in statistically heterogeneous mixtures,” J. Quant. Spectrosc. Radiat. Transfer 36, 557–561 (1986).
[Crossref]

Weaver, W. R.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[Crossref]

Welsh, B. M.

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[Crossref]

Wong, J.

C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
[Crossref]

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

Zumofen, G.

J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
[Crossref]

Appl. Opt. (2)

Bull. Am. Meteorol. Soc. (1)

W. B. Rossow, R. A. Schiffer, “ISCCP cloud data products,” Bull. Am. Meteorol. Soc. 72, 2–20 (1991).
[Crossref]

Eos (1)

B. R. Barkstrom, E. F. Harrison, R. B. Lee, the ERBE Science Team, “Climate and the Earth’s radiation budget,” Eos 71, 297–304 (1990).
[Crossref]

Fractals (1)

J. Klafter, G. Zumofen, M. F. Shlesinger, “Fractal description of anomalous diffusion in dynamical systems,” Fractals 1, 111–126 (1993).
[Crossref]

J. Atmos. Sci. (2)

G. L. Stephens, “Radiative transfer through arbitrarily shaped optical media. Part II: group theory and simple closures,” J. Atmos. Sci. 45, 1837–1848 (1988).
[Crossref]

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[Crossref]

J. Fluid Mech. (3)

K. R. Sreenivasan, C. Meneveau, “The fractal facets of turbulence,” J. Fluid Mech. 173, 357–386 (1986).
[Crossref]

R. R. Prasad, K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (1990).
[Crossref]

G. F. Lane-Serff, “Investigation of the fractal structure of jets and plumes,” J. Fluid Mech. 249, 521–534 (1993).
[Crossref]

J. Geophys. Res. (1)

V. R. Taylor, L. L. Stowe, “Reflectance characteristics of uniform Earth and cloud surfaces derived from NIMBUS-7 ERB,” J. Geophys. Res. 89, 4987–4996 (1984).
[Crossref]

J. Math. Phys. (2)

C. D. Levermore, G. C. Pomraning, D. L. Sanzo, J. Wong, “Linear transport theory in a random medium,” J. Math. Phys. 27, 2526–2536 (1986).
[Crossref]

C. D. Levermore, J. Wong, G. C. Pomraning, “Renewal theory for transport processes in binary statistical mixtures,” J. Math. Phys. 29, 995–1004 (1988).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (5)

D. Vanderhaegen, “Impact of a mixing structure on radiative transfer in random media,” J. Quant. Spectrosc. Radiat. Transfer 39, 333–337 (1988).
[Crossref]

D. Vanderhaegen, “Radiative transfer in statistically heterogeneous mixtures,” J. Quant. Spectrosc. Radiat. Transfer 36, 557–561 (1986).
[Crossref]

W. M. Irvine, “Diffuse reelection and transmission by clouds and dust layers,” J. Quant. Spectrosc. Radiat. Transfer 8, 471–485 (1968).
[Crossref]

G. C. Pomraning, “A model for interface intensities in stochastic particle transport,” J. Quant. Spectrosc. Radiat. Transfer 46, 221–236 (1991).
[Crossref]

G. C. Pomraning, “Radiative transfer in random media with scattering,” J. Quant. Spectrosc. Radiat. Transfer 40, 479–487 (1988).
[Crossref]

Opt. Eng. (2)

J. E. Craig, C. Allen, “Aero-optical turbulent boundary layer/shear layer experiment on the KC-135 aircraft revisited,” Opt. Eng. 24, 446–454 (1985).

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[Crossref]

Phys. Fluids A (1)

P. L. Miller, P. E. Dimotakis, “Stochastic geometric properties of scalar interfaces in turbulent jets,” Phys. Fluids A 3, 168–177 (1991).
[Crossref]

Phys. Rev. Lett. (1)

P. Constantin, I. Procaccia, K. R. Sreenivasan, “Fractal geometry of isoscalar surfaces in turbulence: theory and experiments,” Phys. Rev. Lett. 67, 1739–1742 (1991).
[Crossref] [PubMed]

Physica D (1)

P. Flohr, D. Olivari, “Fractal and multifractal characteristics of a scalar dispersed in a turbulent jet,” Physica D 76, 278–290 (1994).
[Crossref]

Science (2)

S. Lovejoy, “Area-perimeter relation for rain and cloud areas,” Science 216, 185–187 (1982).
[Crossref] [PubMed]

V. Ramanathan, R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom, E. Ahmad, D. Hartmann, “Cloud-radiative forcing and climate: results from the earth radiation budget experiment,” Science 243, 57–63 (1989).
[Crossref] [PubMed]

Transp. Theory Stat. (1)

G. C. Pomraning, “Classic transport problems in binary homogeneous markov statistical mixtures,” Transp. Theory Stat. 17, 595–613 (1988).
[Crossref]

Other (7)

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas and Application (Academic, New York, 1980).

P. Lévy, Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, Paris, 1937).

B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Random Variables (Addison-Wesley, Cambridge, 1954), p. 162.

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

D. Mihalas, Stellar Atmospheres, 2nd ed. (Freeman, San Francisco, Calif., 1978).

F. H. Shu, The Physics of Astrophysics, Vol. 1. Radiation (University Science, Mill Valley, Calif., 1991).

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Figures (6)

Fig. 1
Fig. 1

Semilog plot of photon deposition profiles for a single-scattering albedo equal to 0.95. Depth is measured in units of photon median free path μ. (a) Monte Carlo solution for an exponential free-path distribution. The superposed solid curve shows the Eddington solution. (b) Monte Carlo solution for an α = 1/3 power-law free-path distribution.

Fig. 2
Fig. 2

Log–Log plot of photon deposition profiles obtained from Monte Carlo runs with an α = 1/3 power-law photon free-path probability distribution. Histograms for albedos 0.15, 0.8, and 0.95 (solid, dashed, and dotted–dashed curves) are plotted.

Fig. 3
Fig. 3

Photon deposition (a) exponent and (b) coefficient obtained from α = 1/3 Monte Carlo runs, plotted against the value of single-scattering albedo a. The solid curves are a nonlinear least-squares fit to Eq. (20). The estimated standard deviation of the errors in the Monte Carlo calculated values for δ1/3 and γ1/3 is ~10% of the measured means.

Fig. 4
Fig. 4

Variation of the layer coefficients of absorption, transmission, and reflection with layer thickness, plotted for single-scattering albedo values 0.2, 0.6, 0.9, and 0.99. The layer thickness is measured relative to photon median free path μ. Curves for α = 1/3 power-law (solid curve) and exponential (dashed curve) photon free-path statistics are shown.

Fig. 5
Fig. 5

Semilog plot of photon deposition profiles for a single-scattering albedo equal to 0.95. Depth is measured in units of photon median free path μ. Monte Carlo solutions for an α = 3 power-law free-path distribution (solid curve) and for an exponential distribution (dotted curve), are compared with the theoretical Eddington approximation solution (dashed–dotted curve).

Fig. 6
Fig. 6

Solid curves show the variation of the layer coefficients of absorption, transmission, and reflection with layer thickness for α = 1/4, 1/3, 1/2, and 2/3 at a fixed single-scattering albedo value of 0.8; dashed curves show corresponding results for exponential photon free-path statistics. The layer thickness is measured relative to photon median free path μ.

Equations (28)

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p ( τ n ) = f ( τ n ) τ n 1 + α ,             0 < α ,
x 2 const { t 2 , 0 < α < 1 t 3 - α , 1 < α < 2 t , 2 < α ,
N n = 1 n a n = a ( 1 - a ) 2 ,             0 < a < 1 ,
E ( z ) = e p N dep ( z ) E 0 z - 1 - α ,             0 z ,             0 < α < 1 ,
τ n = i δ t ,
p ( i ) = 1 ζ ( 1 + α ) i 1 + α ,             i = 1 p ( i ) = 1 ,
F 0 = 0 , F i = 1 ζ ( 1 + α ) n = 1 i 1 n 1 + α ,             i = 1 , 2 , 3 , , 2 16 , F 2 16 + 1 = 1 ,
F i - 1 < χ < F i .
1 ζ ( 4 / 3 ) i = 1 4 1 i 4 / 3 = 0.496 ,
c δ t = δ r ,
p ( i ) = [ exp ( 1 / b ) - 1 ] exp ( - i / b ) ,             i = 1 p ( i ) = 1 ,
F 0 = 0 , F i = [ exp ( 1 / b ) - 1 ] n = 1 i exp ( - n / b ) ,             i = 1 , 2 , 3 , , 2 16 , F 2 16 + 1 = 1 ,
[ exp ( 1 / b ) - 1 ] n = 1 4 exp ( - n / b ) = 1 ζ ( 4 / 3 ) n = 1 4 1 n 4 / 3 .
p 1 / 3 ( x ) = { 1.06268 - 1.29062 x , x 0.470508 1 6 x 4 / 3 , 0.470508 x .
F 1 / 3 ( x ) = { 1.06268 x - 0.645311 x 2 , x 0.470508 1 - 1 2 x 1 / 3 , 0.470508 x .
F e ( x ) = 1 - exp [ - ( ln 2 ) x ] .
Δ N d Δ z = σ a σ s ( 15 σ - 10 3 σ a σ ) ( 4 σ a - 3 σ ) ( 3 σ a - σ ) N 0 exp ( - 3 σ a σ z ) + 2 σ a σ ( 3 σ a - σ ) N 0 exp ( - σ z ) ,
σ = 1 a l = ln 2 a μ , σ a = ( 1 - a ) a l = ( 1 - a ) ln 2 a μ , σ s = 1 l = ln 2 μ .
μ Δ N d N 0 Δ z γ α ( μ z ) 1 + δ α ,             0 < δ α α ,
δ 1 / 3 = α ( 1 - a v ) ,             γ 1 / 3 = g ( 1 - a v )
p 3 ( x ) = { 0.51986 - 0.144136 x , x 2.88539 7.20667 x 4 , 2.88539 x ,
F 3 ( x ) = { 0.51986 x - 0.072068 x 2 , x 2.88539 1 - 2.40222 x 3 , 2.88539 x .
F 1 / 4 ( x ) = { 1.63655 x - 1.859930 x 2 , x 0.244416 1 - 1 2 x 1 / 4 , 0.244416 x ,
F 1 / 2 ( x ) = { 0.758519 x - 0.258908 x 2 , x 0.878906 1 - 1 2 x 1 / 2 , 0.878906 x ,
F 2 / 3 ( x ) = { 0.683052 x - 0.18225 x 2 , x 1.17121 1 - 1 2 x 2 / 3 , 1.17121 x
D n ( r ) [ n ( r o + r ) - n ( r o ) ] 2 C m r 2 ,
D s ( r ) = 2 ( 2 π / λ ) 2 0 z ( z - ζ ) [ D n ( ζ 2 + r 2 ) - D n ( ζ ) ] d ζ = ( 2 π / λ ) 2 C m z 2 r 2 ,
P ¯ ( k ) = exp [ - ( 1 / 2 ) D s ( λ f k ) ] = exp [ - 2 π 2 C m z 2 f 2 k 2 ] ,

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