Abstract

The power spectrum of water optical turbulence is shown to vary with its average temperature $\langle T \rangle$ and average salinity concentration $\langle S \rangle$, as well as with light wavelength $\lambda$. This study explores such variations for $\langle T \rangle \in [{0{^ \circ}{\rm C},30{^ \circ}{\rm C}}]$, $\langle S \rangle \in [{0\;{\rm ppt},40\;{\rm ppt}}]$ covering most of the possible natural water conditions within the Earth’s boundary layer and for visible electromagnetic spectrum, $\lambda \in [{400\;{\rm nm},700\;{\rm nm}}]$. For illustration of the effects of these parameters on propagating light, we apply the developed power spectrum model for estimation of the scintillation index of a plane wave (the Rytov variance) and the threshold between weak and strong turbulence regimes.

© 2020 Optical Society of America

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References

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  1. O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2018), Vol. 64, pp. 1–43.
  2. S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).
  3. O. Korotkova, Random Beams: Theory and Applications (CRC Press, 2013).
  4. According to the results of Ref. [5], the refractive index of still oceanic water is a nonlinear function of temperature and salinity concentration. This implies that the fluctuations in the refractive index are not only related with the fluctuations of temperature and salinity but also change with their average values. Besides, most of the fluid parameters in the power spectrum of refractive-index fluctuations directly depend on the average temperature and average salinity. Thus, the optical signal must also be affected by the water’s average temperature and salinity. Similar effect is also pertinent to the air’s refractive index, as it is also a function of average temperature, but such dependence is very small as compared to the water’s refractive index.
  5. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34, 3477–3480 (1995).
    [Crossref]
  6. Water dispersion was addressed in the refractive index model of Ref. [5]. It predicts how the refractive index changes with light wavelength, among other parameters, and hence, affects the propagating light. The power spectrum that we developed includes this dispersion-like dependence and, hence, results in the effects of underwater medium on the statistics of propagating light. The effect of the light wavelength on the laser beam scintillation index was experimentally demonstrated in Ref. [7].
  7. M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
    [Crossref]
  8. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [Crossref]
  9. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
    [Crossref]
  10. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).
    [Crossref]
  11. B. Ruddick and T. Shirtcliffe, “Data for double diffusers: physical properties of aqueous salt-sugar solutions,” Deep Sea Res. Part A 26, 775–787 (1979).
    [Crossref]
  12. 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]
  13. 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]
  14. 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]
  15. J. Yao, H. Zhang, R. Wang, J. Cai, Y. Zhang, and O. Korotkova, “Wide-range Prandtl/Schmidt number power spectrum of optical turbulence and its application to oceanic light propagation,” Opt. Express 27, 27807–27819 (2019).
    [Crossref]
  16. O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. 460, 125119 (2020).
    [Crossref]
  17. R. M. Pope and E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [Crossref]
  18. A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. 13, 265–280 (1983).
    [Crossref]
  19. As described in Ref. [12], eddy diffusivity ratio ${d_r}$dr is not equal to 1, and it is a piecewise function of density ratio ${R_\rho}$Rρ, which varies with thermal expansion coefficient $\alpha$α and saline contraction coefficient $\beta$β. Here we use TEOS-10 toolbox to calculate $\alpha$α and $\beta$β varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩. In combining with ${d_r}({R_\rho})$dr(Rρ), we will get details about ${d_r}$dr varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩.
  20. T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).
  21. T. J. McDougall and P. M. Barker, “Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox,” SCOR/IAPSO WG 127 (2011), pp. 1–28.
  22. P. R. Jackson and C. R. Rehmann, “Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow,” J. Phys. Oceanogr. 33, 1592–1603 (2003).
    [Crossref]
  23. G. Sager, “Zur refraktion von licht im meerwasser,” Beitr. Meeresk. 33, 63–72 (1974).
  24. R. W. Austin and G. Halikas, “The index of refraction of seawater,” Technical Report SIO Ref. No. 76-1 (Defense Advanced Research Projects Agency, 1976).
  25. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  26. The change of refractive index with temperature and salinity is nonlinear, which means the linear coefficients of temperature and salinity should vary with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩, and as we know, refractive index changes with wavelength. For the above reasons, we treat the linear coefficients $A$A and $B$B as environment-dependent. We have derived this in Section 4. This result constitutes the key discrepancy with traditional models like [13] and [15].
  27. Reference [15] gives a spectrum model based on Hill’s model 4, and it has been shown to have a better precision than Nikishov and Nikishov’s model. Here we choose the H4-based spectrum as a basic model for each of the three spectra: the temperature, the salinity, and their co-spectrum.
  28. L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).
  29. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
    [Crossref]
  30. The positive correlation between the plane wave scintillation and $\langle T\rangle$⟨T⟩ is a new result different from previous reports [15,16]. The difference comes from our extended consideration of $A$A, $B$B, and ${d_r}$dr varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩.
  31. D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
    [Crossref]
  32. K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienghard, “Thermophysical properties of seawater: a review and new correlations that include pressure dependence,” Desalination 390, 1–24 (2016).
    [Crossref]
  33. D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination 8, 393–401 (1970).
    [Crossref]
  34. M. H. Sharqawy, J. H. Lienhard, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalin. Water Treat. 16, 354–380 (2010).
    [Crossref]
  35. F. J. Millero, “Seawater as a multicomponent electrolyte solution,” in Marine Chemistry (Wiley, 1974), Vol. 5, pp. 3–80.
  36. J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination 10, 329–339 (1972).
    [Crossref]
  37. F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep Sea Res. Part A 28, 625–629 (1981).
    [Crossref]

2020 (1)

O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. 460, 125119 (2020).
[Crossref]

2019 (1)

2018 (2)

M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
[Crossref]

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]

2017 (2)

2016 (1)

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

2012 (1)

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

2010 (1)

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

2003 (1)

P. R. Jackson and C. R. Rehmann, “Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow,” J. Phys. Oceanogr. 33, 1592–1603 (2003).
[Crossref]

2000 (1)

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

1997 (1)

1995 (1)

1983 (1)

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

1981 (1)

F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep Sea Res. Part A 28, 625–629 (1981).
[Crossref]

1979 (1)

B. Ruddick and T. Shirtcliffe, “Data for double diffusers: physical properties of aqueous salt-sugar solutions,” Deep Sea Res. Part A 26, 775–787 (1979).
[Crossref]

1978 (2)

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

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

1974 (1)

G. Sager, “Zur refraktion von licht im meerwasser,” Beitr. Meeresk. 33, 63–72 (1974).

1972 (1)

J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination 10, 329–339 (1972).
[Crossref]

1970 (1)

D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination 8, 393–401 (1970).
[Crossref]

1969 (1)

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

Abdallah, M.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).

Austin, R. W.

R. W. Austin and G. Halikas, “The index of refraction of seawater,” Technical Report SIO Ref. No. 76-1 (Defense Advanced Research Projects Agency, 1976).

Avramov-Zamurovic, S.

M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
[Crossref]

Banchik, L. D.

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

Barker, P. M.

T. J. McDougall and P. M. Barker, “Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox,” SCOR/IAPSO WG 127 (2011), pp. 1–28.

Baykal, Y.

Cai, J.

Cartwright, G.

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

Chen, C.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Djordjevic, I. B.

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]

Feistel, R.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Fry, E. S.

Halikas, G.

R. W. Austin and G. Halikas, “The index of refraction of seawater,” Technical Report SIO Ref. No. 76-1 (Defense Advanced Research Projects Agency, 1976).

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

Isdale, J.

J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination 10, 329–339 (1972).
[Crossref]

Jackett, D.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Jackson, P. R.

P. R. Jackson and C. R. Rehmann, “Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow,” J. Phys. Oceanogr. 33, 1592–1603 (2003).
[Crossref]

Jamieson, D.

D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination 8, 393–401 (1970).
[Crossref]

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

Kelly, M.

M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
[Crossref]

King, B.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Korotkova, O.

O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. 460, 125119 (2020).
[Crossref]

J. Yao, H. Zhang, R. Wang, J. Cai, Y. Zhang, and O. Korotkova, “Wide-range Prandtl/Schmidt number power spectrum of optical turbulence and its application to oceanic light propagation,” Opt. Express 27, 27807–27819 (2019).
[Crossref]

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

O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2018), Vol. 64, pp. 1–43.

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

Lienghard, J. H.

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

Lienhard, J. H.

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

Marion, G.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

McDougall, T.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

McDougall, T. J.

T. J. McDougall and P. M. Barker, “Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox,” SCOR/IAPSO WG 127 (2011), pp. 1–28.

Millero, F.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Millero, F. J.

F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep Sea Res. Part A 28, 625–629 (1981).
[Crossref]

F. J. Millero, “Seawater as a multicomponent electrolyte solution,” in Marine Chemistry (Wiley, 1974), Vol. 5, pp. 3–80.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

Morris, R.

J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination 10, 329–339 (1972).
[Crossref]

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

Nayar, K. G.

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

Nelson, C.

M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
[Crossref]

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. 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,” Int. J. 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 in Random Media (SPIE, 2005).

Poisson, A.

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

F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep Sea Res. Part A 28, 625–629 (1981).
[Crossref]

Pope, R. M.

Qaraqe, K.

Quan, X.

Rehmann, C. R.

P. R. Jackson and C. R. Rehmann, “Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow,” J. Phys. Oceanogr. 33, 1592–1603 (2003).
[Crossref]

Ruddick, B.

B. Ruddick and T. Shirtcliffe, “Data for double diffusers: physical properties of aqueous salt-sugar solutions,” Deep Sea Res. Part A 26, 775–787 (1979).
[Crossref]

Sager, G.

G. Sager, “Zur refraktion von licht im meerwasser,” Beitr. Meeresk. 33, 63–72 (1974).

Seitz, S.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Sharqawy, M. H.

K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienghard, “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, and S. M. Zubair, “Thermophysical properties of seawater: a review of existing correlations and data,” Desalin. 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]

Shirtcliffe, T.

B. Ruddick and T. Shirtcliffe, “Data for double diffusers: physical properties of aqueous salt-sugar solutions,” Deep Sea Res. Part A 26, 775–787 (1979).
[Crossref]

Spitzer, P.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Thorpe, S. A.

S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).

Tudhope, J.

D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination 8, 393–401 (1970).
[Crossref]

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

Uysal, M.

Wang, R.

Wang, X.

Wang, Y.

Wright, D.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

Yao, J.

Yao, J.-R.

O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. 460, 125119 (2020).
[Crossref]

Yi, X.

Zhang, H.

Zhang, Y.

Zubair, S. M.

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

Appl. Opt. (2)

Beitr. Meeresk. (1)

G. Sager, “Zur refraktion von licht im meerwasser,” Beitr. Meeresk. 33, 63–72 (1974).

Deep Sea Res. Part A (2)

F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep Sea Res. Part A 28, 625–629 (1981).
[Crossref]

B. Ruddick and T. Shirtcliffe, “Data for double diffusers: physical properties of aqueous salt-sugar solutions,” Deep Sea Res. Part A 26, 775–787 (1979).
[Crossref]

Desalin. Water Treat. (1)

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

Desalination (4)

J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination 10, 329–339 (1972).
[Crossref]

D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,” Desalination 7, 23–30 (1969).
[Crossref]

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

D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination 8, 393–401 (1970).
[Crossref]

Int. J. Fluid Mech. Res. (1)

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

J. Fluid Mech. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. Oceanogr. (1)

P. R. Jackson and C. R. Rehmann, “Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow,” J. Phys. Oceanogr. 33, 1592–1603 (2003).
[Crossref]

Mar. Chem. (1)

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

Opt. Commun. (1)

O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. 460, 125119 (2020).
[Crossref]

Opt. Express (3)

Proc. SPIE (1)

M. Kelly, S. Avramov-Zamurovic, and C. Nelson, “Exploration of multiple wavelength laser beams propagating underwater,” Proc. SPIE 10631, 1063118 (2018).
[Crossref]

Waves Random Complex Media (1)

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

Other (15)

The positive correlation between the plane wave scintillation and $\langle T\rangle$⟨T⟩ is a new result different from previous reports [15,16]. The difference comes from our extended consideration of $A$A, $B$B, and ${d_r}$dr varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩.

F. J. Millero, “Seawater as a multicomponent electrolyte solution,” in Marine Chemistry (Wiley, 1974), Vol. 5, pp. 3–80.

R. W. Austin and G. Halikas, “The index of refraction of seawater,” Technical Report SIO Ref. No. 76-1 (Defense Advanced Research Projects Agency, 1976).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

The change of refractive index with temperature and salinity is nonlinear, which means the linear coefficients of temperature and salinity should vary with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩, and as we know, refractive index changes with wavelength. For the above reasons, we treat the linear coefficients $A$A and $B$B as environment-dependent. We have derived this in Section 4. This result constitutes the key discrepancy with traditional models like [13] and [15].

Reference [15] gives a spectrum model based on Hill’s model 4, and it has been shown to have a better precision than Nikishov and Nikishov’s model. Here we choose the H4-based spectrum as a basic model for each of the three spectra: the temperature, the salinity, and their co-spectrum.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).

As described in Ref. [12], eddy diffusivity ratio ${d_r}$dr is not equal to 1, and it is a piecewise function of density ratio ${R_\rho}$Rρ, which varies with thermal expansion coefficient $\alpha$α and saline contraction coefficient $\beta$β. Here we use TEOS-10 toolbox to calculate $\alpha$α and $\beta$β varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩. In combining with ${d_r}({R_\rho})$dr(Rρ), we will get details about ${d_r}$dr varying with ${\langle}T \rangle $⟨T⟩ and ${\langle}S \rangle $⟨S⟩.

T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz, “The international thermodynamic equation of seawater 2010 (TEOS-10): calculation and use of thermodynamic properties,” Global Ship-based Repeat Hydrography Manual, IOCCP Report No 14 (2009).

T. J. McDougall and P. M. Barker, “Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox,” SCOR/IAPSO WG 127 (2011), pp. 1–28.

Water dispersion was addressed in the refractive index model of Ref. [5]. It predicts how the refractive index changes with light wavelength, among other parameters, and hence, affects the propagating light. The power spectrum that we developed includes this dispersion-like dependence and, hence, results in the effects of underwater medium on the statistics of propagating light. The effect of the light wavelength on the laser beam scintillation index was experimentally demonstrated in Ref. [7].

O. Korotkova, “Light propagation in a turbulent ocean,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2018), Vol. 64, pp. 1–43.

S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007).

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

According to the results of Ref. [5], the refractive index of still oceanic water is a nonlinear function of temperature and salinity concentration. This implies that the fluctuations in the refractive index are not only related with the fluctuations of temperature and salinity but also change with their average values. Besides, most of the fluid parameters in the power spectrum of refractive-index fluctuations directly depend on the average temperature and average salinity. Thus, the optical signal must also be affected by the water’s average temperature and salinity. Similar effect is also pertinent to the air’s refractive index, as it is also a function of average temperature, but such dependence is very small as compared to the water’s refractive index.

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Figures (7)

Fig. 1.
Fig. 1. (a) Schmidt number and (b) Prandtl number varying with average temperature and salinity.
Fig. 2.
Fig. 2. ${d_r}$ varying with $\langle T \rangle$ and $\langle S \rangle$ at different values of $H$. (a) $H = - {10^ \circ}{\rm C} \cdot {{\rm ppt}^{- 1}}$; (b) $H = - {100^ \circ}{\rm C} \cdot {{\rm ppt}^{- 1}}$; (c) $H = - {400^ \circ}{\rm C} \cdot {{\rm ppt}^{- 1}}$.
Fig. 3.
Fig. 3. Coefficients $A$ and $B$ varying with $\langle T\rangle$ and $\langle S\rangle$ at $\lambda = 532\; {\rm nm}$.
Fig. 4.
Fig. 4. $A$ and $B$ varying with $\lambda$ when $\langle T\rangle = 15^ \circ{\rm C},\langle S\rangle = 34.9\;{\rm ppt}$.
Fig. 5.
Fig. 5. (a) $n^\prime $ varying with $T^\prime $, and (b) $n^\prime $ varying with $S^\prime $ in three models: Model 1, Nikishov and Nikishov’s model; Model 2, our linear model in Eq. (15) with Eqs. (16)–(17); Model 3, Quan and Fry’s formula containing full polynomial.
Fig. 6.
Fig. 6. Scintillation index $\sigma _{I,{\rm pl}}^2(L)$ with different values of (a) average salinity, (b) average temperature, and (c) wavelength.
Fig. 7.
Fig. 7. $\log ({{L_d}})$ varying with $\langle T \rangle$ and $\langle S \rangle$.

Equations (36)

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P r = ν α T ,
ν = μ ρ ,
α T = σ T ρ c p ,
P r = c p μ σ T .
S c = ν α S ,
α S μ T = c o n s t a n t .
α S 5.954 × 10 15 T μ .
S c = μ α S ρ μ 2 5.954 × 10 15 T ρ .
d r = K S / K T { R ρ + R ρ 0.5 ( R ρ 1 ) 0.5 , R ρ 1 , 1.85 R ρ 0.85 , 0.5 R ρ < 1 , 0.15 R ρ , R ρ < 0.5 ,
R ρ = α | H | β ,
α ( T , S ) = 1 V V T | S a n d β ( T , S ) = 1 V V S | T ,
n ( T , S , λ ) = a 0 + ( a 1 + a 2 T + a 3 T 2 ) S + a 4 T 2 + a 5 + a 6 S + a 7 T λ + a 8 λ 2 + a 9 λ 3 ,
a 0 = 1.31405 , a 1 = 1.779 × 10 4 , a 2 = 1.05 × 10 6 , a 3 = 1.6 × 10 8 , a 4 = 2.02 × 10 6 , a 5 = 15.868 , a 6 = 0.01155 , a 7 = 0.00423 , a 8 = 4382 , a 9 = 1.1455 × 10 6 .
n = n 0 ( T , S , λ ) + n ,
n A ( T , S , λ ) T + B ( T , S , λ ) S ,
d n ( T , S , λ ) = n ( T , S , λ ) T d T + n ( T , S , λ ) S d S + n ( T , S , λ ) λ d λ .
A ( T , S , λ ) = n ( T , S , λ ) T | T = T , S = S = a 2 S + 2 a 3 T S + 2 a 4 T + a 7 λ ,
B ( T , S , λ ) = n ( T , S , λ ) S | T = T , S = S = a 1 + a 2 T + a 3 T 2 + a 6 λ .
Φ n ( κ , T , S , λ ) = A 2 ( T , S , λ ) Φ T ( κ ) + B 2 ( T , S , λ ) Φ S ( κ ) + 2 A ( T , S , λ ) B ( T , S , λ ) Φ T S ( κ ) ,
Φ i ( κ , T , S ) = [ 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 0 ε 1 3 κ 11 3 χ i exp [ 176.90 ( κ η ) 2 c i 0 .96 ] , i { T , S , T S } ,
{ χ T = K T ( d T d z ) 2 , χ S = K S ( d S d z ) 2 , χ TS = K T + K S 2 ( d T d z ) ( d S d z ) ,
η = ν 3 / 4 ε 1 / 4 = [ μ ( T , S ) ρ ( T , S ) ] 3 / 4 ε 1 / 4 ,
{ c T = 0.072 4 / 3 β P r 1 ( T , S ) , c S = 0.072 4 / 3 β S c 1 ( T , S ) , c TS = 0.072 4 / 3 β P r ( T , S ) + S c ( T , S ) 2 P r ( T , S ) S c ( T , S ) .
{ χ S ( T , S , H , χ T ) = d r ( T , S , H ) H 2 χ T , χ TS ( T , S , H , χ T ) = 1 + d r ( T , S , H ) 2 H χ T ,
σ I , p l 2 ( L ) = 8 π 2 k 0 2 L n 0 2 0 1 d ξ 0 κ [ 1 cos ( L κ 2 ξ k 0 ) ] × Φ n ( κ , T , S , λ ) d κ = 8 π 2 k 0 2 L n 0 2 0 κ [ 1 sin ( L κ 2 / k 0 ) L κ 2 / k 0 ] × Φ n ( κ , T , S , λ ) d κ ,
σ I , p l 2 ( L d ) = 1 ,
c p = 1000 × ( a 11 + a 12 T + a 13 T 2 + a 14 T 2 ) ,
{ a 11 = 5.328 9.76 × 10 2 S + 4.04 × 10 4 S 2 , a 12 = 6.913 × 10 3 + 7.351 × 10 4 S 3.15 × 10 6 S 2 , a 13 = 9.6 × 10 6 1.927 × 10 6 S + 8.23 × 10 9 S 2 , a 14 = 2.5 × 10 9 + 1.666 × 10 9 S 7.125 × 10 12 S 2 .
log ( σ T ) = log ( 240 + 0.0002 S h ) 3 + 0.434 × ( 2.3 343.5 + 0.037 S h T h + 273.15 ) × [ 1 T h + 273.15 647.3 + 0.03 S h ] 1 / 3 ,
T h = 1.00024 T a n d S h = S / 1.00472.
{ μ = μ 0 ( a 21 s + a 22 s 2 ) , s = S × 10 3 ,
a 21 = 1.5409136040 + 1.9981117208 × 10 2 T 9.5203865864 × 10 5 T 2 , a 22 = 7.9739318223 7.5614568881 × 10 2 T + 4.7237011074 × 10 4 T 2 ,
μ 0 = [ 0.15700386464 × ( T + 64.992620050 ) 2 91.296496657 ] 1 + 4.2844324477 × 10 5 .
ρ = ρ T + ρ S ,
ρ T = 9.9992293295 × 10 2 + 2.0341179217 × 10 2 T 6.1624591598 × 10 3 T 2 + 2.2614664708 × 10 5 T 3 4.6570659168 × 10 8 T 4 ,
ρ TS = s [ 8.0200240891 × 10 2 2.0005183488 T + 1.6771024982 × 10 2 T 2 3.0600536746 × 10 5 T 3 1.6132224742 × 10 5 T 2 s ] ,

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