Abstract

The analytical formulae for the wave structure functions (WSF) and the spatial coherence radius of plane and spherical waves propagating through oceanic turbulence are derived. It is found that the Kolmogorov five-thirds power law of WSF is also valid for oceanic turbulence in the inertial range. The changes of the WSF and the spatial coherence radius versus different parameters of oceanic turbulence are examined.

© 2014 Optical Society of America

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References

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  2. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, Vol. XXII, E. Wolf, ed. (Elsevier, 1985), Chap. VI.
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    [Crossref]
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    [Crossref]
  10. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
    [Crossref]
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  15. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
    [Crossref]
  16. M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
    [Crossref]
  17. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014).
    [Crossref] [PubMed]
  18. Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves in Random Complex Media 24(2), 164–173 (2014).
    [Crossref]
  19. L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
    [Crossref]
  20. L. C. Andrews, Special Functions of Mathematics for Engineers, 3rd ed. (SPIE and Oxford University, 1998)
  21. R. M. Pope and E. S. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36(33), 8710–8723 (1997).
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    [Crossref]

2014 (2)

Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014).
[Crossref] [PubMed]

Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves in Random Complex Media 24(2), 164–173 (2014).
[Crossref]

2013 (2)

2012 (3)

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

2011 (2)

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[Crossref]

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

2009 (2)

2006 (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

2005 (1)

2003 (1)

2002 (1)

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

1997 (1)

1993 (1)

L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
[Crossref]

1992 (1)

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Andrews, L. C.

L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
[Crossref]

Arnon, S.

Ata, Y.

Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014).
[Crossref] [PubMed]

Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves in Random Complex Media 24(2), 164–173 (2014).
[Crossref]

Baykal, Y.

Davidson, F. M.

Farwell, N.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[Crossref]

Flatley, J. P.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Freeman, D. E.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Fry, E. S.

Gbur, G.

Goode, W.

Hou, W.

Jarosz, E.

Ji, G. M.

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Ji, X. L.

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Kedar, D.

Korotkova, O.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[Crossref]

Landry, M. A.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Li, X. Q.

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Lindstrom, C. E.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Liu, L.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Longacre, J. R.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Pope, R. M.

Richardson, C. E.

L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
[Crossref]

Ricklin, J. C.

Schwartz, J. A.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
[Crossref]

Snow, J. B.

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Tang, M.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

Vester, S.

L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
[Crossref]

Weidemann, A.

Wolf, E.

Woods, S.

Zhao, D.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111(4), 665–670 (2013).
[Crossref]

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

V. V. Nikishov and V. I. Nikishov, “Spectum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

J. Mod. Opt. (1)

L. C. Andrews, S. Vester, and C. E. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40(5), 931–938 (1993).
[Crossref]

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

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

J. Opt. Soc. Am. A (5)

New J. Phys. (1)

X. L. Ji, X. Q. Li, and G. M. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Opt. Commun. (2)

O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011).
[Crossref]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

J. B. Snow, J. P. Flatley, D. E. Freeman, M. A. Landry, C. E. Lindstrom, J. R. Longacre, and J. A. Schwartz, “Underwater propagation of high-data rate laser communications pulses,” Proc. SPIE 1750, 419–427 (1992).
[Crossref]

Waves in Random Complex Media (1)

Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves in Random Complex Media 24(2), 164–173 (2014).
[Crossref]

Waves in Random Complex Media. (1)

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves in Random Complex Media. 22(2), 260–266 (2012).
[Crossref]

Other (3)

L. C. Andrews, Special Functions of Mathematics for Engineers, 3rd ed. (SPIE and Oxford University, 1998)

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

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, Vol. XXII, E. Wolf, ed. (Elsevier, 1985), Chap. VI.

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Figures (6)

Fig. 1
Fig. 1 Curves of WSF versus w .
Fig. 2
Fig. 2 Curves of WSF versus log χ T .
Fig. 3
Fig. 3 Curves of WSF versus logε .
Fig. 4
Fig. 4 Changes of ρ 0 versus w .
Fig. 5
Fig. 5 Changes of ρ 0 versus log χ T .
Fig. 6
Fig. 6 Changes of ρ 0 versus logε .

Equations (26)

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Φ n ( κ )=0.388× 10 8 ε 1 /3 κ 11 /3 [1+2.35 (κη) 2/3 ] χ T w 2 ( w 2 e A T δ + e A S δ 2w e A TS δ ),
D pl (ρ,L)=8 π 2 k 2 L 0 [1 J 0 (κρ) ] Φ n (κ)κdκ,
D pl (ρ,L)=A n=1 (1) n1 ρ 2n (n!) 2 2 2n 0 κ 2n 8 3 [1+g κ 2 3 ]( w 2 e a κ 4 3 b κ 2 + e c κ 4 3 d κ 2 2w e e κ 4 3 f κ 2 )dκ,
0 κ 2n 8 3 e Q κ 4 3 R κ 2 dκ = 1 4 R 1 2 n {2 R 4 3 Γ(n 5 6 ) F 2 2 ( n 2 5 12 , n 2 + 1 12 ; 1 3 , 2 3 ; 4 Q 3 27 R 2 ) 2Q R 2 3 Γ(n 1 6 ) F 2 2 ( n 2 1 12 , n 2 + 5 12 ; 2 3 , 4 3 ; 4 Q 3 27 R 2 ) + Q 2 Γ(n+ 1 2 ) F 2 2 ( n 2 + 1 4 , n 2 + 3 4 ; 4 3 , 5 3 ; 4 Q 3 27 R 2 )},
0 κ 2n2 e Q κ 4 3 R κ 2 dκ = 1 4 R 5 6 n {2 R 4 3 Γ(n 1 2 ) F 2 2 ( n 2 1 4 , n 2 + 1 4 ; 1 3 , 2 3 ; 4 Q 3 27 R 2 ) 2Q R 2 3 Γ(n+ 1 6 ) F 2 2 ( n 2 + 1 12 , n 2 + 7 12 ; 2 3 , 4 3 ; 4 Q 3 27 R 2 ) + Q 2 Γ(n+ 5 6 ) F 2 2 ( n 2 + 5 12 , n 2 + 11 12 ; 4 3 , 5 3 ; 4 Q 3 27 R 2 )},
F 2 2 (α,β;γ,ς;x){ 1 αβx γς , | x |1 Γ(γ)Γ(ς)Γ(βα) x α Γ(β)Γ(γα)Γ(ςα) ,Re(x)1 .
D pl (ρ,L)A{ 1 2 Γ( 5 6 ) b 5 6 w 2 (1 7 a 3 216 b 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4b )] + 1 2 Γ( 5 6 ) d 5 6 (1 7 c 3 216 d 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4d )] Γ( 5 6 ) f 5 6 w(1 7 e 3 216 f 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4f )] + 5 12 Γ( 5 6 )a b 1 6 g w 2 (1 91 a 3 864 b 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4b )] + 5 12 Γ( 5 6 )c d 1 6 g(1 91 c 3 864 d 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4d )] 5 6 Γ( 5 6 )e f 1 6 gw(1 91 e 3 864 f 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4f )] + 1 2 Γ( 1 2 ) b 1 2 g w 2 (1 a 3 8 b 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4b )] + 1 2 Γ( 1 2 ) d 1 2 g(1 c 3 8 d 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4d )] Γ( 1 2 ) f 1 2 gw(1 e 3 8 f 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4f )] 1 8 Γ( 1 2 ) a 2 b 1 2 w 2 (1 a 3 16 b 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4b )] 1 8 Γ( 1 2 ) c 2 d 1 2 (1 c 3 16 d 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4d )] + 1 4 Γ( 1 2 ) e 2 f 1 2 w(1 e 3 16 f 2 )[1 F 1 1 ( 1 2 ;1; ρ 2 4f )] 1 2 Γ( 1 6 )a b 1 6 w 2 (1 55 a 3 864 b 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4b )] 1 2 Γ( 1 6 )c d 1 6 (1 55 c 3 864 d 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4d )] +Γ( 1 6 )e f 1 6 w(1 55 e 3 864 f 2 )[1 F 1 1 ( 1 6 ;1; ρ 2 4f )] 1 24 Γ( 1 6 ) a 2 b 5 6 g w 2 (1 187 a 3 2160 b 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4b )] 1 24 Γ( 1 6 ) c 2 d 5 6 g(1 187 c 3 2160 d 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4d )] + 1 12 Γ( 1 6 ) e 2 f 5 6 gw(1 187 e 3 2160 f 2 )[1 F 1 1 ( 5 6 ;1; ρ 2 4f )]}.
F 1 1 (α;β;x){ 1 αx β , | x |1 Γ(β) Γ(βα) x α , Re(x)1 ,
D pl (ρ,L){ 3.603× 10 7 k 2 L ε 1/3 χ T w 2 ρ 2 (16.958 w 2 44.175w+118.923),(ρη) 3.603× 10 7 k 2 L ε 1/3 χ T w 2 ρ 5/3 (1.116 w 2 2.235w+1.119), (ρη) .
ρ 0pl { [3.603× 10 7 k 2 L ε 1/3 χ T 2 w 2 (16.958 w 2 44.175w+118.923)] 1 /2 ,( ρ 0 η) [3.603× 10 7 k 2 L ε 1/3 χ T 2 w 2 (1.116 w 2 2.235w+1.119)] 3 /5 , ( ρ 0 η) .
D sp (ρ,L)=8 π 2 k 2 L 0 1 0 [1 J 0 (κξρ) ] Φ n (κ)κdκdξ ,
D sp (ρ,L){ 3.603× 10 7 k 2 L ε 1/3 χ T w 2 ρ 2 (5.623 w 2 14.725w+39.641),(ρη) 3.603× 10 7 k 2 L ε 1/3 χ T w 2 ρ 5/3 (0.419 w 2 0.838w+0.419), (ρη) ,
ρ 0sp { [3.603× 10 7 k 2 L ε 1/3 χ T 2 w 2 (5.623 w 2 14.725w+39.641)] 1 /2 ,( ρ 0 η) [3.603× 10 7 k 2 L ε 1/3 χ T 2 w 2 (0.419 w 2 0.838w+0.419)] 3 /5 ,( ρ 0 η) .
D(ρ,L)={ 2 (ρ/ ρ 0 ) 2 , (ρη) 2 (ρ/ ρ 0 ) 5/3 , (ρη) .
E n (κ)= C 0 χ n ε 1 /3 κ 5 /3 [ 1+ C 1 (κη) 2/3 ] × w 2 θexp( A T δ)+exp( A S δ)w(1+θ)exp( A TS δ) w 2 θ+1w(1+θ) ,
E n (κ)= C 0 ( α 2 χ T + α 2 w 2 χ T 2 α 2 w χ T ) ε 1 /3 κ 5 /3 [ 1+ C 1 (κη) 2/3 ] × w 2 θexp( A T δ)+exp( A S δ)w(1+θ)exp( A TS δ) w 2 θ+1w(1+θ) .
E n (κ)= C 0 α 2 ε 1 /3 κ 5 /3 [ 1+ C 1 (κη) 2/3 ] χ T w 2 ×[ w 2 exp( A T δ)+exp( A S δ)2wexp( A TS δ)].
Φ n (κ)= (4π) 1 κ 2 E n (κ) =0.388× 10 8 ε 1 /3 κ 11 /3 [ 1+2.35 (κη) 2/3 ] χ T w 2 ×[ w 2 exp( A T δ)+exp( A S δ)2wexp( A TS δ)].
χ T = K T ( d T 0 dz ) 2 ,
w= α(d T 0 /dz) β(d S 0 /dz) ,
Φ n (κ)=0.388× 10 8 ε 1 /3 κ 11 /3 [1+2.35 (κη) 2/3 ]{ K T ( d T 0 dz ) 2 e A T δ +{ K T ( d T 0 dz ) 2 / [ α(d T 0 /dz) β(d S 0 /dz) ] 2 } e A S δ [ 2 K T ( d T 0 dz ) 2 / α(d T 0 /dz) β(d S 0 /dz) ] e A TS δ } =0.388× 10 8 ε 1 /3 κ 11 /3 [1+2.35 (κη) 2/3 ]{ K T (d T 0 /dz) 2 e A T δ + K T [β(d S 0 /dz)/α] 2 e A S δ [2 K T β(d T 0 /dz)(d S 0 /dz)/α] e A TS δ }.
Φ n (κ)=0.388× 10 8 ε 1 /3 κ 11 /3 [1+2.35 (κη) 2/3 ] χ S (β/α) 2 e A S δ ,
D pl (ρ,L){ 4.285× 10 5 k 2 L ε 1/3 χ S (β/α) 2 ρ 2 ,(ρη) 4.032× 10 7 k 2 L ε 1/3 χ S (β/α) 2 ρ 5/3 , (ρη) ,
ρ 0pl { {2.142× 10 5 k 2 L ε 1/3 χ S (β/α) 2 } 1 /2 ,( ρ 0 η) {2.016× 10 7 k 2 L ε 1/3 χ S (β/α) 2 } 3 /5 , ( ρ 0 η) .
D sp (ρ,L){ 1.428× 10 5 k 2 L ε 1/3 χ S (β/α) 2 ρ 2 ,(ρη) 1.510× 10 7 k 2 L ε 1/3 χ S (β/α) 2 ρ 5/3 , (ρη) ,
ρ 0sp { [0.714× 10 5 k 2 L ε 1/3 χ S (β/α) 2 ] 1 /2 ,( ρ 0 η) [0.755× 10 7 k 2 L ε 1/3 χ S (β/α) 2 ] 3 /5 ,( ρ 0 η) .

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