Abstract

Light influenced by the turbulent ocean can be fully characterized with the help of the power spectrum of the water’s refractive index fluctuations, resulting from the combined effect of two scalars, temperature and salinity concentration advected by the velocity field. The Nikishovs’ model [ Fluid Mech. Res. 27, 8298 (2000)] frequently used in the analysis of light evolution through the turbulent ocean channels is the linear combination of the temperature spectrum, the salinity spectrum and their co-spectrum, each being described by an approximate expression developed by Hill [ J. Fluid Mech. 88, 541562 (1978)] in the first of his four suggested models. The fourth of the Hill’s models provides much more precise power spectrum than the first one expressed via a non-linear differential equation that does not have a closed-form solution. We develop an accurate analytic approximation to the fourth Hill’s model valid for Prandtl/Schmidt numbers in the interval [3, 3000] and use it for the development of a more precise oceanic power spectrum. To illustrate the advantage of our model, we include numerical examples relating to the spherical wave scintillation index evolving in the underwater turbulent channels with different average temperatures, and, hence, different Prandtl numbers for temperature and different Schmidt numbers for salinity. Since our model is valid for a large range of Prandtl number (or/and Schmidt number), it can be readily adjusted to oceanic waters with seasonal or extreme average temperature and/or salinity or any other turbulent fluid with one or several advected quantities.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Power spectrum of refractive-index fluctuations in turbulent ocean and its effect on optical scintillation

Xiang Yi and Ivan B. Djordjevic
Opt. Express 26(8) 10188-10202 (2018)

Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation

R. J. Hill and S. F. Clifford
J. Opt. Soc. Am. 68(7) 892-899 (1978)

Investigation of Hill’s optical turbulence model by means of direct numerical simulation

Andreas Muschinski and Stephen M. de Bruyn Kops
J. Opt. Soc. Am. A 32(12) 2423-2430 (2015)

References

  • View by:
  • |
  • |
  • |

  1. W. Hou, “A simple underwater imaging model,” Opt. Lett. 34, 2688–2690 (2009).
    [Crossref] [PubMed]
  2. X. Yi, Z. Li, and Z. Liu, “Underwater optical communication performance for laser beam propagation through weak oceanic turbulence,” Appl. Opt. 54, 1273–1278 (2015).
    [Crossref] [PubMed]
  3. Y. Baykal, “Bit error rate of pulse position modulated optical wireless communication links in oceanic turbulence,” J. Opt. Soc. Am. A 35, 1627–1632 (2018).
    [Crossref]
  4. Z. Cui, P. Yue, X. Yi, and J. Li, “Scintillation of a partially coherent beam with pointing errors resulting from a slightly skewed underwater platform in oceanic turbulence,” Appl. Opt. 58, 4443–4449 (2019).
    [Crossref] [PubMed]
  5. O. Korotkova, “Enhanced backscatter in lidar systems with retro-reflectors operating through a turbulent ocean,” J. Opt. Soc. Am. A 35, 1797–1804 (2018).
    [Crossref]
  6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).
    [Crossref]
  7. O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2013).
  8. O. O. Chumak and R. A. Baskov, “Strong enhancing effect of correlations of photon trajectories on laser beam scintillations,” Phys. Rev. A 93, 033821 (2016).
    [Crossref]
  9. R. A. Baskov and O. O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
    [Crossref]
  10. O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, vol. 64T. D. Visser, ed. (Elsevier, 2019), pp. 1–43.
    [Crossref]
  11. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Fluid Mech. Res. 27, 82–98 (2000).
    [Crossref]
  12. L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
    [Crossref]
  13. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
    [Crossref]
  14. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011).
    [Crossref]
  15. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
    [Crossref]
  16. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
    [Crossref]
  17. Y. Baykal, “Scintillation index in strong oceanic turbulence,” Opt. Commun. 375, 15–18 (2016).
    [Crossref]
  18. Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves Random Complex Media 24, 164–173 (2014).
    [Crossref]
  19. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22, 27112–27122 (2014).
    [Crossref] [PubMed]
  20. J. Yao, Y. Zhang, R. Wang, Y. Wang, and X. Wang, “Practical approximation of the oceanic refractive index spectrum,” Opt. Express 25, 23283–23292 (2017).
    [Crossref] [PubMed]
  21. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
    [Crossref]
  22. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
    [Crossref]
  23. X. Yi and I. B. Djordjevic, “Power spectrum of refractive-index fluctuations in turbulent ocean and its effect on optical scintillation,” Opt. Express 26, 10188–10202 (2018).
    [Crossref] [PubMed]
  24. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [Crossref]
  25. Further, H4 agrees with the Kraichnan spectrum at high wave numbers, while H1 evidently deviates from it. The precision of the Kraichnan spectrum and, hence, of H4 is well established [26–29].
  26. A. Muschinski and S. M. de Bruyn Kops, “Investigation of hill’s optical turbulence model by means of direct numerical simulation,” J. Opt. Soc. Am. A 32, 2423–2430 (2015).
    [Crossref]
  27. D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
    [Crossref]
  28. X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
    [Crossref]
  29. D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
    [Crossref]
  30. Y. Li, Y. Zhang, and Y. Zhu, “Oceanic spectrum of unstable stratification turbulence with outer scale and scintillation index of gaussian-beam wave,” Opt. Express 27, 7656–7672 (2019).
    [Crossref] [PubMed]
  31. J. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” Opt. Acta Int. J. Opt. 37, 13–16 (1990).
  32. R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmospheric Sci. 49, 1494–1509 (1992).
    [Crossref]
  33. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” Opt. Acta Int. J. Opt. 39, 1849–1853 (1992).
  34. K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
    [Crossref]
  35. A. Muschinski, “Temperature variance dissipation equation and its relevance for optical turbulence modeling,” J. Opt. Soc. Am. A 32, 2195–2200 (2015).
    [Crossref]
  36. The value of β may change with environment, and has always been used as an experimental constant. Several experiments gave its range 0.22 ∼ 0.78 [39]. The value 0.72 has been widely used in oceanic optical research, so we use the value 0.72 in Table 1 and Figure 2, but keep it as a parameter in all formulae.
  37. P. Yue, X. Luan, X. Yi, Z. Cui, and M. Wu, “Beam-wander analysis in turbulent ocean with the effect of the eddy diffusivity ratio and the outer scale,” J. Opt. Soc. Am. A 36, 556–562 (2019).
    [Crossref]
  38. The Gaussian tail is used here for historical reasons and mathematical convenience. Some well-known power spectrum models developed by Tatarskii [6] and Andrews [33] rely on the Gaussian profile of the high frequency cut-off. Such Gaussian profile is crucial for convergence of integrals involved in evaluation in statistics of light field propagation.
  39. H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
    [Crossref]
  40. We separate (κη) and c in order to raise them to different powers. The applied mathematical recipe is to make the maximum value of g(κη) vary with c, and thus with Pr. Such variation is the key feature of H4 model [29]. If not separated, c and Pr only scale the curve of g(κη) horizontally but do not change its maximum value.
  41. M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwater optical scintillation index,” J. Opt. Soc. Am. A 34, 1969–1973 (2017).
    [Crossref]
  42. Y. L. Keke, The special functions and their approximations (Academic, 1969).
  43. K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
    [Crossref]
  44. M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
    [Crossref]
  45. A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
    [Crossref]

2019 (3)

2018 (4)

2017 (2)

2016 (3)

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

Y. Baykal, “Scintillation index in strong oceanic turbulence,” Opt. Commun. 375, 15–18 (2016).
[Crossref]

O. O. Chumak and R. A. Baskov, “Strong enhancing effect of correlations of photon trajectories on laser beam scintillations,” Phys. Rev. A 93, 033821 (2016).
[Crossref]

2015 (3)

2014 (2)

2012 (3)

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[Crossref]

D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
[Crossref]

2011 (3)

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011).
[Crossref]

2010 (1)

M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
[Crossref]

2009 (1)

2007 (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
[Crossref]

2006 (1)

L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
[Crossref]

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

1997 (1)

D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[Crossref]

1992 (2)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmospheric Sci. 49, 1494–1509 (1992).
[Crossref]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” Opt. Acta Int. J. Opt. 39, 1849–1853 (1992).

1990 (1)

J. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” Opt. Acta Int. J. Opt. 37, 13–16 (1990).

1983 (1)

A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
[Crossref]

1978 (3)

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

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
[Crossref]

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
[Crossref]

1968 (1)

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

Abdallah, M.

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” Opt. Acta Int. J. Opt. 39, 1849–1853 (1992).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).
[Crossref]

Ata, Y.

Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves Random Complex Media 24, 164–173 (2014).
[Crossref]

Banchik, L. D.

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

Baskov, R. A.

R. A. Baskov and O. O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

O. O. Chumak and R. A. Baskov, “Strong enhancing effect of correlations of photon trajectories on laser beam scintillations,” Phys. Rev. A 93, 033821 (2016).
[Crossref]

Baykal, Y.

Bogucki, D.

D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[Crossref]

Bogucki, D. J.

D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
[Crossref]

Chumak, O. O.

R. A. Baskov and O. O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

O. O. Chumak and R. A. Baskov, “Strong enhancing effect of correlations of photon trajectories on laser beam scintillations,” Phys. Rev. A 93, 033821 (2016).
[Crossref]

Churnside, J.

J. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” Opt. Acta Int. J. Opt. 37, 13–16 (1990).

Cui, Z.

de Bruyn Kops, S. M.

Djordjevic, I. B.

Domaradzki, J. A.

D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
[Crossref]

D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[Crossref]

Elamassie, M.

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011).
[Crossref]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
[Crossref]

Forcat, F.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

Frehlich, R.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmospheric Sci. 49, 1494–1509 (1992).
[Crossref]

Grant, H. L.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
[Crossref]

Hill, R. J.

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

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
[Crossref]

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
[Crossref]

Hou, W.

Hughes, B. A.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

Ji, X.

Keke, Y. L.

Y. L. Keke, The special functions and their approximations (Academic, 1969).

Korotkova, O.

O. Korotkova, “Enhanced backscatter in lidar systems with retro-reflectors operating through a turbulent ocean,” J. Opt. Soc. Am. A 35, 1797–1804 (2018).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[Crossref]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011).
[Crossref]

O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2013).

O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, vol. 64T. D. Visser, ed. (Elsevier, 2019), pp. 1–43.
[Crossref]

Li, J.

Li, Y.

Li, Z.

Lienhard V, J. H.

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
[Crossref]

Liu, L.

L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
[Crossref]

Liu, Z.

Lu, L.

Luan, X.

Luo, H.

D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
[Crossref]

Moilliet, A.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

Muschinski, A.

Nayar, K. G.

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

Papaud, A.

A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).
[Crossref]

Planella, J.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

Poisson, A.

A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
[Crossref]

Qaraqe, K.

Roget, E.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

Sanchez, X.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

Sharqawy, M. H.

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
[Crossref]

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[Crossref]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
[Crossref]

Sun, J.

L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
[Crossref]

Uysal, M.

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
[Crossref]

Vogel, W. M.

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

Wang, R.

Wang, X.

Wang, Y.

Wei, L.

L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
[Crossref]

Wu, M.

Yao, J.

Yeung, P. K.

D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[Crossref]

Yi, X.

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
[Crossref]

Yue, P.

Zhang, Y.

Zhu, Y.

Zubair, S. M.

M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415 (2011).
[Crossref]

Desalination (1)

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysical properties of seawater: A review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
[Crossref]

Desalination Water Treat. (1)

M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalination Water Treat. 16, 354–380 (2010).
[Crossref]

Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

J. Atmospheric Sci. (1)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmospheric Sci. 49, 1494–1509 (1992).
[Crossref]

J. Fluid Mech. (3)

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

H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, “The spectrum of temperature fluctuations in turbulent flow,” J. Fluid Mech. 34, 423–442 (1968).
[Crossref]

D. Bogucki, J. A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with pr>1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[Crossref]

J. Opt. A-Pure Appl. Opt. (1)

L. Wei, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A-Pure Appl. Opt. 8, 1052 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Phys. Oceanogr. (2)

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: The batchelor and kraichnan models,” J. Phys. Oceanogr. 41, 2155–2167 (2011).
[Crossref]

D. J. Bogucki, H. Luo, and J. A. Domaradzki, “Experimental evidence of the kraichnan scalar spectrum at high reynolds numbers,” J. Phys. Oceanogr. 42, 1717–1728 (2012).
[Crossref]

Mar. Chem. (1)

A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
[Crossref]

Opt. Acta Int. J. Opt. (2)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” Opt. Acta Int. J. Opt. 39, 1849–1853 (1992).

J. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” Opt. Acta Int. J. Opt. 37, 13–16 (1990).

Opt. Commun. (3)

Y. Baykal, “Scintillation index in strong oceanic turbulence,” Opt. Commun. 375, 15–18 (2016).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (2)

O. O. Chumak and R. A. Baskov, “Strong enhancing effect of correlations of photon trajectories on laser beam scintillations,” Phys. Rev. A 93, 033821 (2016).
[Crossref]

R. A. Baskov and O. O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

Radio Sci. (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13, 953–961 (1978).
[Crossref]

Waves Random Complex Media (3)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18, 173–184 (2007).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[Crossref]

Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves Random Complex Media 24, 164–173 (2014).
[Crossref]

Other (8)

O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, vol. 64T. D. Visser, ed. (Elsevier, 2019), pp. 1–43.
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).
[Crossref]

O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2013).

The value of β may change with environment, and has always been used as an experimental constant. Several experiments gave its range 0.22 ∼ 0.78 [39]. The value 0.72 has been widely used in oceanic optical research, so we use the value 0.72 in Table 1 and Figure 2, but keep it as a parameter in all formulae.

The Gaussian tail is used here for historical reasons and mathematical convenience. Some well-known power spectrum models developed by Tatarskii [6] and Andrews [33] rely on the Gaussian profile of the high frequency cut-off. Such Gaussian profile is crucial for convergence of integrals involved in evaluation in statistics of light field propagation.

We separate (κη) and c in order to raise them to different powers. The applied mathematical recipe is to make the maximum value of g(κη) vary with c, and thus with Pr. Such variation is the key feature of H4 model [29]. If not separated, c and Pr only scale the curve of g(κη) horizontally but do not change its maximum value.

Y. L. Keke, The special functions and their approximations (Academic, 1969).

Further, H4 agrees with the Kraichnan spectrum at high wave numbers, while H1 evidently deviates from it. The precision of the Kraichnan spectrum and, hence, of H4 is well established [26–29].

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 Universal dimensionless function g(κη) at different Prandtl numbers: (a) Pr=7, (b) Pr=700.
Fig. 2
Fig. 2 Universal dimensionless function g(κη) at different Prandtl numbers: (a) Pr = 3, (b) Pr = 30, (c) Pr = 300, (d) Pr = 3000.
Fig. 3
Fig. 3 The values of X vary with Prandtl number in H1 model, H4 model and Eq. (5).
Fig. 4
Fig. 4 The scintillation index of a spherical wave corresponding to different power spectrum models, for several different values of ω and 〈T〉: (a) ω = −0.25, 〈T〉 = 0°C, (b) ω = −5, 〈T〉 = 0°C, (c) ω = −0.25, 〈T〉 = 15°C, (d) ω = −5, 〈T〉 = 15°C, (e) ω = −0.25, 〈T〉 = 30°C, (f) ω = −5, 〈T〉 = 30°C. a Curve 1 (red, dashed curve) is calculated from Hill’s model 4 numerically. b Curve 2 (black, solid curve) is based on Eq. (16). c Curve 3 (blue, dotted curve) is calculated from Nikishovs’ spectrum numerically.

Tables (3)

Tables Icon

Table 1 Values of X calculated from Eq. (5) for various Prandtl numbers.

Tables Icon

Table 2 The values of X corresponding to reported models

Tables Icon

Table 3 Values of viscosity, diffusivity and Prandtl/Schmidt number in different values of temperature when salinity is 34.9ppt

Equations (22)

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

Φ n ( κ ) = 1 4 π β χ ε 1 3 κ 11 3 g ( κ η ) ,
d d z { ( z 2 b + 1 ) 1 3 b [ 11 3 f ( z ) + z d f ( z ) d z ] } = 22 3 c z 1 3 f ( z ) ,
f ( z ) = g ( a z ) = g ( κ η ) ,
g ( κ η ) = [ 1 + h 1 ( κ η ) h 2 c h 3 + h 4 ( κ η ) h 5 c h 6 ] × exp [ h 7 ( κ η ) 2 c h 8 ] ,
g ( κ η ) = [ 1 + 21.61 ( κ η ) 0.61 c 0.02 18.18 ( κ η ) 0.55 c 0.04 ] × exp [ 174.90 ( κ η ) 2 c 0.96 ] .
Φ n ( κ ) = [ 1 + 21.61 ( κ η ) 0.61 c 0.02 18.18 ( κ η ) 0.55 c 0.04 ] × 1 4 π β ε 1 3 κ 11 3 χ exp [ 174.90 ( κ η ) 2 c 0.96 ] .
X = 2 β Pr 0 g ( x ) x 1 3 d x .
n = AT + BS
Φ n ( κ ) = A 2 Φ T ( κ ) + B 2 Φ S ( κ ) 2 A B Φ TS ( κ ) .
Φ i ( κ ) = [ 1 + 21.61 ( κ η ) 0.61 c i 0.02 18.18 ( κ η ) 0.55 c i 0.04 ] × 1 4 π β ε 1 3 κ 11 3 χ i exp [ 174.90 ( κ η ) 2 c i 0.96 ] , i { T , S , TS } ,
χ S = A 2 ω 2 B 2 χ T d r , χ TS = A 2 ω B χ T ( 1 + d r ) ,
d r { | ω | + | ω | 0.5 ( | ω | 1 ) 0.5 , | ω | 1 1.85 | ω | 0.85 , 0.5 | ω | < 1 0.15 | ω | , | ω | < 0.5 .
σ I 2 ( L ) = I 2 ( L ) I ( L ) 2 1 ,
σ I 2 ( L ) = 4 π Re { 0 L d ς 0 κ d κ 0 2 π d θ [ | E ( ς , κ , L ) | 2 + E ( ς , κ , L ) × E ( ς , κ , L ) ] Φ n ( κ ) n 0 2 } ,
E ( ς , κ , L ) = i k exp [ 0.5 i ς ( L ς ) k L κ 2 ] ,
σ I 2 ( L ) = ( A 2 χ T M T + B 2 χ S M S 2 A B χ TS M TS ) × π L β ε 1 3 η 5 3 ( k n 0 ) 2 ,
M i = = 1 3 P , 1 c i P , 2 ( 174.90 c i 0.96 ) ( 5 6 P , 3 2 ) Γ ( P , 3 2 5 6 ) × [ 1 F 2 3 ( 1 , P , 3 4 5 12 , P , 3 4 1 12 ; 3 4 , 5 4 ; L 2 16 × ( 174.90 c i 0.96 ) 2 k 2 η 4 ) ] , { i = T , S , TS }
P = { P i j } = ( 1 0 0 21.61 0.02 0.61 18.18 0.04 0.55 ) .
Φ ( κ ) = 1 4 π β χ ε 1 3 κ 11 3 g ( κ η ) ,
g ( k η ) = [ 1 + Q ( k η ) 2 3 ] exp [ β 3 2 Q 2 ( k η ) 4 3 + Q 3 ( k η ) 2 Q 2 Pr ] .
Φ n ( κ ) = A 2 Φ T ( κ ) + B 2 Φ S ( κ ) 2 A B Φ TS ( κ ) ,
Φ i ( κ ) = 1 4 π β ε 1 3 χ i κ 11 3 [ 1 + Q ( κ η ) 2 3 ] × exp [ β 3 2 Q 2 ( κ η ) 4 / 3 + Q 3 ( κ η ) 2 Q 2 Pr i ] , { i = T , S , TS } .

Metrics