Abstract

Oceanic turbulence is described by the oceanic refractive-index spectrum (ORIS), which considers several important hydrodynamic parameters. Based on ORIS, many optical oceanic quantities can be calculated using numerical integration. However, it is difficult to calculate the analytical solutions. In this paper, an approximate oceanic temperature spectrum is obtained by multiplying the non-Kolmogorov spectrum with a correction factor. By analogy with the obtained temperature spectrum, an approximate salinity spectrum and an approximate coupling spectrum are obtained. A linear summation of these three approximate spectra forms the approximate form of ORIS. The approximate form of ORIS we obtained helps calculate the analytical solutions of the relevant oceanic optical quantities.

© 2017 Optical Society of America

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References

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2017 (1)

2016 (8)

2015 (4)

2014 (5)

2013 (1)

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

2012 (2)

Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B-Lasers Opt. 109(2), 289–294 (2012).
[Crossref]

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

Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36(23), 4554–4556 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

L. B. Liu, S. L. Zhou, and J. H. Cui, “Prospects and problems of wireless communication for underwater sensor networks,” Wirel. Commun. Mob. Comput. 8(8), 977–994 (2008).
[Crossref]

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

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(1), 82–98 (2000).
[Crossref]

1978 (1)

1977 (1)

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(03), 423–442 (1968).
[Crossref]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

Arnon, S.

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[Crossref]

Ata, Y.

Baykal, Y.

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref] [PubMed]

Y. Baykal, “Fourth-order mutual coherence function in oceanic turbulence,” Appl. Opt. 55(11), 2976–2979 (2016).
[Crossref] [PubMed]

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

M. C. Gökçe and Y. Baykal, “Scintillation analysis of multiple-input single-output underwater optical links,” Appl. Opt. 55(22), 6130–6136 (2016).
[Crossref] [PubMed]

Y. Baykal, “Scintillations of LED sources in oceanic turbulence,” Appl. Opt. 55(31), 8860–8863 (2016).
[Crossref] [PubMed]

Y. Baykal, “Expressing oceanic turbulence parameters by atmospheric turbulence structure constant,” Appl. Opt. 55(6), 1228–1231 (2016).
[Crossref] [PubMed]

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, “Field correlation of spherical wave in underwater turbulent medium,” Appl. Opt. 53(33), 7968–7971 (2014).
[Crossref] [PubMed]

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(22), 27112–27122 (2014).
[Crossref] [PubMed]

Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36(23), 4554–4556 (2011).
[Crossref] [PubMed]

Bissonnette, L. R.

Cai, Y.

Chen, Q.

Y. Zhou, Q. Chen, and D. M. Zhao, “Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence,” Appl. Phys. B-Lasers Opt. 114(4), 475–482 (2014).
[Crossref]

Cheng, J.

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[Crossref]

Clifford, S.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).

Cui, J. H.

L. B. Liu, S. L. Zhou, and J. H. Cui, “Prospects and problems of wireless communication for underwater sensor networks,” Wirel. Commun. Mob. Comput. 8(8), 977–994 (2008).
[Crossref]

Cui, L.

Du, W.

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves 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]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

Gerçekcioglu, H.

Ghassemlooy, Z.

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[Crossref]

Gökçe, M. C.

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(03), 423–442 (1968).
[Crossref]

Han, Q.

Hill, R.

Huang, K. K.

Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B-Lasers Opt. 109(2), 289–294 (2012).
[Crossref]

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(03), 423–442 (1968).
[Crossref]

Ji, X.

Korotkova, O.

X. Xiao, D. G. Voelz, I. Toselli, and O. Korotkova, “Gaussian beam propagation in anisotropic turbulence along horizontal links: theory, simulation, and laboratory implementation,” Appl. Opt. 55(15), 4079–4084 (2016).
[Crossref] [PubMed]

O. Korotkova, “Polarization changes in light beams trespassing anisotropic turbulence,” Opt. Lett. 40(13), 3077–3080 (2015).
[Crossref] [PubMed]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves 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]

Kulikov, V. A.

V. A. Kulikov, “Estimation of turbulent parameters based on the intensity scintillations of the laser beam propagated through a turbulent water layer,” J. Appl. Phys. 119(12), 123103 (2016).
[Crossref]

Li, Y.

Liu, L.

Liu, L. B.

L. B. Liu, S. L. Zhou, and J. H. Cui, “Prospects and problems of wireless communication for underwater sensor networks,” Wirel. Commun. Mob. Comput. 8(8), 977–994 (2008).
[Crossref]

Lu, L.

Ma, J.

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(03), 423–442 (1968).
[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(1), 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(1), 82–98 (2000).
[Crossref]

Peng, X.

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

Rao, R.

Shchepakina, E.

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

Tan, L.

Tang, M. M.

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

Toselli, I.

X. Xiao, D. G. Voelz, I. Toselli, and O. Korotkova, “Gaussian beam propagation in anisotropic turbulence along horizontal links: theory, simulation, and laboratory implementation,” Appl. Opt. 55(15), 4079–4084 (2016).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

Uysal, M.

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[Crossref]

Voelz, D. G.

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(03), 423–442 (1968).
[Crossref]

Wu, X.

Wu, Y.

Xiao, X.

Xu, Z.

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[Crossref]

Yu, S.

Zhang, Y.

Zhao, D. M.

Y. Zhou, Q. Chen, and D. M. Zhao, “Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence,” Appl. Phys. B-Lasers Opt. 114(4), 475–482 (2014).
[Crossref]

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

Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B-Lasers Opt. 109(2), 289–294 (2012).
[Crossref]

Zhou, S. L.

L. B. Liu, S. L. Zhou, and J. H. Cui, “Prospects and problems of wireless communication for underwater sensor networks,” Wirel. Commun. Mob. Comput. 8(8), 977–994 (2008).
[Crossref]

Zhou, Y.

Y. Zhou, Q. Chen, and D. M. Zhao, “Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence,” Appl. Phys. B-Lasers Opt. 114(4), 475–482 (2014).
[Crossref]

Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B-Lasers Opt. 109(2), 289–294 (2012).
[Crossref]

Zhu, W.

Zhu, Y.

Appl. Opt. (7)

Appl. Phys. B-Lasers Opt. (3)

Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B-Lasers Opt. 109(2), 289–294 (2012).
[Crossref]

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

Y. Zhou, Q. Chen, and D. M. Zhao, “Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence,” Appl. Phys. B-Lasers Opt. 114(4), 475–482 (2014).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging Optical Wireless Communications-Advances and Challenges,” IEEE J. Sel. Areas Comm. 33(9), 1738–1749 (2015).
[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(1), 82–98 (2000).
[Crossref]

J. Appl. Phys. (1)

V. A. Kulikov, “Estimation of turbulent parameters based on the intensity scintillations of the laser beam propagated through a turbulent water layer,” J. Appl. Phys. 119(12), 123103 (2016).
[Crossref]

J. Fluid Mech. (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(03), 423–442 (1968).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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]

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

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (2)

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).

Waves Random Complex Media (1)

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

Wirel. Commun. Mob. Comput. (1)

L. B. Liu, S. L. Zhou, and J. H. Cui, “Prospects and problems of wireless communication for underwater sensor networks,” Wirel. Commun. Mob. Comput. 8(8), 977–994 (2008).
[Crossref]

Other (1)

R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE, 2005).

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Figures (4)

Fig. 1
Fig. 1

A plot of f T0 , f T01 , and f T02 , where Pr T =7.

Fig. 2
Fig. 2

The plot of F i ( Pr i ).

Fig. 3
Fig. 3

f i1 ( κη ) and f i0 ( κη ) for different Prandtl numbers. (a) Pr i =5, C 1i =2.165 (b) Pr i =10, C 1i =2.205 (c) Pr i =15, C 1i =2.226 (d) Pr i =600, C 1i =2.315 (e) Pr i =700, C 1i =2.317 (f) Pr i =800, C 1i =2.318

Fig. 4
Fig. 4

The comparison between the 1D-TS predicted by the approximate ORIS and that measured by Grant et al. (a) No.1, (b) No.2, (c) No.3, (d) No.4 aLine 1 is the 1D-TS corresponding to (21). bThe discrete points are 1D-TS measured by Grant et al. [31]. cLine 2 is the fitted curve given by Grant et al. [31].

Tables (1)

Tables Icon

Table 1 The values for Pr T , η and C 1T

Equations (30)

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

Φ T (κ)= C T 2 κ α T exp[ κ 2 / κ mT 2 ], 3< α T <4
Φ T (κ)= C T 2 exp[ κ 2 / κ mT 2 ][ 1+ C 1T (κη) C 2T ] κ α T
Φ n 0 (κ)= A 2 Φ T 0 (κ)+ B 2 Φ S 0 (κ)2AB Φ TS 0 (κ)
Φ i 0 (κ)= (4π) 1 β ε 1/3 η 11/3 [ 1+Q (κη) 2/3 ] G i χ i , i=T,S,TS , G i (κη)= (κη) 11/3 exp( A i δ ) ,δ= 3 2 Q 2 ( κη ) 4/3 + Q 3 ( κη ) 2 , A i =β Q 2 Pr i 1 , Pr TS = 2 Pr T Pr S Pr T + Pr S , χ T = K T ( d T ¯ dz ) 2 , χ S = K S ( d S ¯ dz ) 2 , χ TS = K T + K S 2 ( d T ¯ dz )( d S ¯ dz ).
Φ n 0 (κ)| d S ¯ dz =0 = A 2 Φ T 0 (κ) = A 2 (4π) 1 β ε 1/3 η 11/3 χ T [ 1+ (κη) 2/3 Q ] G T
f T ( ηκ )=4π ε 1/3 χ T 1 κ 11/3 Φ T
f T0 =4π ε 1/3 χ T 1 κ 11/3 Φ T 0 = f T01 + f T02
f T01 =βexp[ A T ( 3 2 Q 2 ( κη ) 4/3 + Q 3 ( κη ) 2 ) ] , f T02 =βQ (κη) 2 3 exp[ A T ( 3 2 Q 2 ( κη ) 4/3 + Q 3 ( κη ) 2 ) ] .
f T1 =4π ε 1/3 χ T 1 κ 11/3 Φ T = f T11 + f T12
f T11 =4π ε 1/3 χ T 1 κ 11/3 C T 2 exp[ κ 2 / κ mT 2 ] κ α T , f T12 =4π ε 1/3 χ T 1 κ 11/3 C T 2 exp[ κ 2 / κ mT 2 ] C 1T (κη) C 2T κ α T .
Φ T (κ) C T 2 [ 1+ C 1T (κη) 2/3 ] κ 11/3 exp[ κ 2 / κ mT 2 ]
C T 2 = (4π) 1 β ε 1/3 χ T
f T11 =βexp( κ 2 κ mT 2 )
f T12 =β C 1T (κη) 2/3 exp( κ 2 κ mT 2 ).
( κη )| d f T02 d( κη ) =0 = ( κη )| d f T12 d( κη ) =0
κ mT = 3 η Q 3/2 ( W T 1 3 + 1 9 W T ) 3/2
W T = { [ ( 1 27 Pr T 6β Q 2 ) 2 1 729 ] 1/2 ( 1 27 Pr T 6β Q 2 ) } 1/3
Φ T ( κ ) C T 2 κ 11/3 exp[ ( κη ) 2 ( κ mT η ) 2 ]+ C 1T C T 2 η 2/3 κ 3 exp[ ( κη ) 2 ( κ mT η ) 2 ]
κ mT η= 3 Q 3/2 ( W T 1 3 + 1 9 W T ) 3/2 , C T 2 = (4π) 1 β ε 1/3 χ T ,
Φ S (κ) C S 2 κ 11/3 exp[ (κη) 2 ( κ mS η) 2 ]+ C 1S C S 2 η 2/3 κ 3 exp[ (κη) 2 ( κ mS η) 2 ]
Φ TS ( κ ) C TS 2 κ 11/3 exp[ (κη) 2 ( κ mTS η) 2 ]+ C 1TS C TS 2 η 2/3 κ 3 exp[ (κη) 2 ( κ mTS η) 2 ]
Φ n (κ)= A 2 Φ T (κ)+ B 2 Φ S (κ)2AB Φ TS (κ),
Φ i (κ)= C i 2 κ 11/3 exp[ (κη) 2 / F i 2 ]+ C i 2 C 1i η 2/3 κ 3 exp[ (κη) 2 / F i 2 ] , C i 2 = (4π) 1 β ε 1/3 χ i , F i = 3 Q 3/2 ( W i 1 3 + 1 9 W i ) 3/2 , W i = { [ ( 1 27 Pr i 6β Q 2 ) 2 1 729 ] 1/2 ( 1 27 Pr i 6β Q 2 ) } 1/3 , i=T,S,TS .
g= 0 dκf(κ) Φ n (κ)
Φ n (κ)=4π ε 1/3 κ 11/3 ( A 2 χ T f T ( ηκ )+ B 2 χ S f S ( ηκ )2AB χ TS f TS ( ηκ ) ).
f i0 =β( 1+Q (κη) 2 3 )exp[ A i ( 3 2 Q 2 ( κη ) 4/3 + Q 3 ( κη ) 2 ) ]
f i1 =β( 1+ C 1i (κη) 2/3 )exp( ( κη ) 2 F i 2 ) i=T,S,TS
ϕ T (κ)= 4π κ 2 A 2 Φ n | χ S =0 . =β ε 1/3 χ T κ 2 exp[ (κη) 2 / F T 2 ]+ C 1T β ε 1/3 χ T η 2/3 κ 4/3 exp[ (κη) 2 / F T 2 ].
η= ε 1/2 / ( α 3/2 k s 3 )
Pr T =ε/ ( α 3 k s 4 ) .