Abstract

We develop a new spectrum of the refractive-index fluctuations for the unstable stratification ocean based on the linear combination of the temperature spectrum, salinity spectrum and coupling spectrum that all include the outer scale. Our oceanic spectrum agrees better with the experimental data than others do from low wave-number regions to high wave-number regions. Based on our proposed oceanic spectrum, we derive the analytical expression of the scintillation index of Gaussian-beam wave and investigate the influence of the light source and the channel parameters on the scintillation index of Gaussian-beam wave.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2018 (4)

2017 (2)

2016 (7)

Z. Wang, P. Zhang, C. Qiao, L. Lu, C. Fan, and X. Ji, “Scintillation index of Gaussian waves in weak turbulent ocean,” Opt. Commun. 380, 79–86 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free space optical communication links with Bessel Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 7901411 (2016).
[Crossref]

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

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

S. A. Arpali, Y. Baykal, and Ç. Arpali, “BER evaluations for multimode beams in underwater turbulence,” J. Mod. Opt. 63(13), 1297–1300 (2016).
[Crossref]

Y. Baykal, “Higher order mode laser beam scintillations in oceanic medium,” Wave Random Complex 26(1), 21–29 (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]

2015 (5)

2014 (3)

2012 (2)

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

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

2011 (2)

L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011).
[Crossref] [PubMed]

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(11), 2155–2167 (2011).
[Crossref]

2009 (2)

2006 (1)

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

2000 (2)

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]

W. J. Merryfield, “Origin of thermohaline staircases,” J. Phys. Oceanogr. 30(5), 1046–1068 (2000).
[Crossref]

1997 (2)

D. J. Bogucki, A. J. 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]

B. L. Ellerbroek, “Including outer scale effects in zonal adaptive optics calculations,” Appl. Opt. 36(36), 9456–9467 (1997).
[Crossref] [PubMed]

1995 (1)

C. Y. Shen, “Equilibrium salt-fingering convection,” Phys. Fluids 7(4), 706–717 (1995).
[Crossref]

1991 (1)

1978 (1)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(3), 541–562 (1978).
[Crossref]

1977 (1)

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

1968 (2)

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]

R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11(5), 945–953 (1968).
[Crossref]

1959 (1)

G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. 5(1), 113–133 (1959).
[Crossref]

Abdallah, M.

Arpali, Ç.

S. A. Arpali, Y. Baykal, and Ç. Arpali, “BER evaluations for multimode beams in underwater turbulence,” J. Mod. Opt. 63(13), 1297–1300 (2016).
[Crossref]

Arpali, S. A.

S. A. Arpali, Y. Baykal, and Ç. Arpali, “BER evaluations for multimode beams in underwater turbulence,” J. Mod. Opt. 63(13), 1297–1300 (2016).
[Crossref]

Ata, Y.

Bai, X.

Batchelor, G. K.

G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. 5(1), 113–133 (1959).
[Crossref]

Baykal, Y.

Y. Ata and Y. Baykal, “Effect of anisotropy on bit error rate for an asymmetrical Gaussian beam in a turbulent ocean,” Appl. Opt. 57(9), 2258–2262 (2018).
[Crossref] [PubMed]

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(11), 1969–1973 (2017).
[Crossref] [PubMed]

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

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]

S. A. Arpali, Y. Baykal, and Ç. Arpali, “BER evaluations for multimode beams in underwater turbulence,” J. Mod. Opt. 63(13), 1297–1300 (2016).
[Crossref]

Y. Baykal, “Higher order mode laser beam scintillations in oceanic medium,” Wave Random Complex 26(1), 21–29 (2016).
[Crossref]

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

Y. Baykal, “Intensity fluctuations of multimode laser beams in underwater medium,” J. Opt. Soc. Am. A 32(4), 593–598 (2015).
[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. 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]

H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009).
[Crossref] [PubMed]

Bogucki, D. J.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

D. J. Bogucki, A. J. 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]

Cai, Y.

Cao, L.

Cao, X.

Champagne, F. H.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Cheng, M.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free space optical communication links with Bessel Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 7901411 (2016).
[Crossref]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref] [PubMed]

Cui, L.

Djordjevic, I. B.

Domaradzki, A. J.

D. J. Bogucki, A. J. 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]

Domaradzki, J. A.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

Elamassie, M.

Ellerbroek, B. L.

Eyyuboglu, H. T.

Fan, C.

Z. Wang, P. Zhang, C. Qiao, L. Lu, C. Fan, and X. Ji, “Scintillation index of Gaussian waves in weak turbulent ocean,” Opt. Commun. 380, 79–86 (2016).
[Crossref]

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012).
[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(11), 2155–2167 (2011).
[Crossref]

Friehe, C. A.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Gerçekcioglu, H.

Gökçe, M. C.

Golmohammady, S.

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]

Gui, Z.

B. Luo, G. Wu, L. Yin, Z. Gui, and Y. Tian, “Propagation of optical coherence lattices in oceanic turbulence,” Opt. Commun. 425, 80–84 (2018).
[Crossref]

Guo, L.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free space optical communication links with Bessel Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 7901411 (2016).
[Crossref]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref] [PubMed]

Hao, L.

Hill, R. J.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(3), 541–562 (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(03), 423–442 (1968).
[Crossref]

Ji, X.

Z. Wang, P. Zhang, C. Qiao, L. Lu, C. Fan, and X. Ji, “Scintillation index of Gaussian waves in weak turbulent ocean,” Opt. Commun. 380, 79–86 (2016).
[Crossref]

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]

Jianning, S.

Jie, M.

Kashani, F. D.

Korotkova, O.

Kraichnan, R.

R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11(5), 945–953 (1968).
[Crossref]

LaRue, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Li, J.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free space optical communication links with Bessel Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 7901411 (2016).
[Crossref]

Li, Z.

Liu, L.

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

Liu, Z.

Lou, H.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

Lu, L.

Z. Wang, P. Zhang, C. Qiao, L. Lu, C. Fan, and X. Ji, “Scintillation index of Gaussian waves in weak turbulent ocean,” Opt. Commun. 380, 79–86 (2016).
[Crossref]

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]

Lu, W.

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

Luo, B.

B. Luo, G. Wu, L. Yin, Z. Gui, and Y. Tian, “Propagation of optical coherence lattices in oceanic turbulence,” Opt. Commun. 425, 80–84 (2018).
[Crossref]

Mashal, A.

Merryfield, W. J.

W. J. Merryfield, “Origin of thermohaline staircases,” J. Phys. Oceanogr. 30(5), 1046–1068 (2000).
[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(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]

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(11), 2155–2167 (2011).
[Crossref]

Qaraqe, K.

Qiao, C.

Z. Wang, P. Zhang, C. Qiao, L. Lu, C. Fan, and X. Ji, “Scintillation index of Gaussian waves in weak turbulent ocean,” Opt. Commun. 380, 79–86 (2016).
[Crossref]

Renmin, Y.

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(11), 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(11), 2155–2167 (2011).
[Crossref]

Sermutlu, E.

Shchepakina, E.

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

Shen, C. Y.

C. Y. Shen, “Equilibrium salt-fingering convection,” Phys. Fluids 7(4), 706–717 (1995).
[Crossref]

Sun, J. F.

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

Tian, Y.

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Figures (6)

Fig. 1
Fig. 1 Comparisons of Kraichnan spectrum, Batchelor spectrum, Yi spectrum, Nikishov spectrum and Our spectrum with (a) experimental data measured by Champagne et al. [29], (b) DNS results [37], and (c) experimental data measured by Grant et al. [30].
Fig. 2
Fig. 2 Spectra of temperature and salinity fluctuations in turbulent flow for different ϖ (a) ϖ=2, (b) ϖ=0.5.
Fig. 3
Fig. 3 Scintillation index as a function of Λ 0 for different outer scale L 0 .
Fig. 4
Fig. 4 Scintillation index as a function of Λ 0 for different (a) rate of dissipation of kinetic energy per unit mass of fluid ε, (b) dissipation rate of temperature variance χ T .
Fig. 5
Fig. 5 Scintillation index versus the kinematic viscosity v for different wavelength λ.
Fig. 6
Fig. 6 Scintillation index as a function of Λ 0 for different the ratio of temperature and salinity contributions to the refractive-index spectrum ϖ and the eddy diffusivity ratio θ.

Tables (2)

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Table 1 Oceanic temperature scalar spectral models

Tables Icon

Table 2 Run parameters of oceanic temperature spectral models

Equations (31)

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n=AT+BS,
Φ n ( κ )= ( 4π κ 2 ) 1 [ A 2 Φ T ( κ )+ B 2 Φ S ( κ )2AB Φ TS ( κ ) ]
Φ ni ( κ )= ε 1/3 β G i ( κ ) [ 1+Q ( κη ) 2/3 ] χ i 4π ( κ 2 + κ 0 2 ) 11/6 ,0<κ<,
χ n = A 2 χ T + B 2 χ S 2AB χ TS .
Φ n ( κ )= ε 1/3 β [ A 2 χ T G T ( κ ) + B 2 χ S G S ( κ )2AB χ TS G TS ( κ ) ] [ 1+ C 1 ( κη ) 2/3 ] 4π ( κ 2 + κ 0 2 ) 11/6 ,
χ T = K T ( d T 0 dz ) 2 , χ S = K S ( d S 0 dz ) 2 , χ TS = K T + K S 2 ( d T 0 dz )( d S 0 dz ),
Φ n ( κ )= β[ 1+ C 1 ( κη ) 2/3 ] ε 1/3 χ n [ ϖ 2 θ+1ϖ( θ+1 ) ]4π ( κ 2 + κ 0 2 ) 11/6 [ G S ( κ )+ ϖ 2 θ G T ( κ )ϖ( 1+θ ) G TS ( κ ) ],
θ= | ϖ | R F { 1/( 1 ( | ϖ |1 )/| ϖ | )| ϖ |1 1.85| ϖ |0.850.5| ϖ |1, 0.15| ϖ || ϖ |0.5
Φ n ( κ )= ε 1/3 β A 2 χ T [ 1+ C 1 ( κη ) 2/3 ] 4π ( κ 2 + κ 0 2 ) 11/6 { exp[ ( κη ) 2 R T 2 ] + 1 ϖ 2 θ exp[ ( κη ) 2 R S 2 ] 1+θ ϖθ exp[ ( κη ) 2 R TS 2 ] },0<κ<.
σ I,oc 2 ( r,L,η, L 0 )= σ I,r 2 ( r,L,η, L 0 )+ σ I,l 2 ( L,η, L 0 ),
σ I,l 2 ( L,η, L 0 )=8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp( ΛL κ 2 ξ 2 k ){ 1cos[ L κ 2 k ξ( 1 Θ ˜ ξ ) ] }dκdξ,
σ I,r 2 ( r,L,η, L 0 )=8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp( ΛL κ 2 ξ 2 k ){ I 0 ( 2Λrκξ )1 }dκdξ,
σ I,l 2 ( L,η, L 0 )=2π k 2 Lβ A 2 ε 1/3 χ T 0 1 0 κ [ 1+ C 1 ( κη ) 2/3 ] ( κ 2 + κ 0 2 ) 11/6 ( { exp[ ( η 2 κ 2 R T 2 + ΛL κ 2 ξ 2 k ) ] + 1 ϖ 2 θ exp[ ( η 2 κ 2 R S 2 + ΛL κ 2 ξ 2 k ) ] 1+θ ϖθ exp[ ( η 2 κ 2 R TS 2 + ΛL κ 2 ξ 2 k ) ] } Re( exp{ [ η 2 κ 2 R T 2 + ΛL κ 2 ξ 2 k +i L κ 2 k ξ( 1 Θ ˜ ξ ) ] }+exp{ [ η 2 κ 2 R S 2 + ΛL κ 2 ξ 2 k + i L κ 2 k ξ( 1 Θ ˜ ξ ) ] }+exp { [ η 2 κ 2 R TS 2 + ΛL κ 2 ξ 2 k +i L κ 2 k ξ( 1 Θ ˜ ξ ) ] } ) )dκdξ.
0 κ 2μ exp( κ 2 / κ m 2 ) ( κ 0 2 + κ 2 ) 11/6 dκ 1 2 κ 0 2μ8/3 Γ( μ+ 1 2 )[ Γ( 4/3μ ) Γ( 11/6 ) + Γ( μ4/3 ) Γ( μ+1/2 ) ( κ 0 2 κ m 2 ) 4 3 μ ],
σ I,l 2 ( L )=π k 2 Lβ A 2 ε 1/3 χ T Re 0 1 { 6.6796( { [ η 2 R T 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 5/6 + 1 ϖ 2 θ [ η 2 R S 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 5/6 ( 1+θ ) ϖθ [ η 2 R TS 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 5/6 } [ ( η 2 R T 2 + ΛL ξ 2 k ) 5/6 + 1 ϖ 2 θ ( η 2 R T 2 + ΛL ξ 2 k ) 5/6 ( 1+θ ) ϖθ ( η 2 R T 2 + ΛL ξ 2 k ) 5/6 ] ) +3.545 C 1 η 2/3 ( { [ η 2 R T 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 1/2 + 1 ϖ 2 θ [ η 2 R S 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 1/2 ( 1+θ ) ϖθ [ η 2 R TS 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] 1/2 } [ ( η 2 R T 2 + ΛL ξ 2 k ) 1/2 + 1 ϖ 2 θ ( η 2 R S 2 + ΛL ξ 2 k ) 1/2 ( 1+θ ) ϖθ ( η 2 R T 2 + ΛL ξ 2 k ) 1/2 ] ) }dξ.
F 2 1 ( a,b;c;z )= Γ( c ) Γ( b )Γ( cb ) 0 1 t b1 ( 1t ) cb1 ( 1tz ) a dt,c>b>0,
0 1 [ η 2 R j 2 + ΛL ξ 2 k +i L k ξ( 1 Θ ˜ ξ ) ] α dξ= ( η R j ) 2α F 2 1 ( α,1;2; iL R j 2 k η 2 + 2 3 ( iΛ+ Θ ˜ ) iL R j 2 k η 2 ),
σ I,l 2 ( L,η, L 0 )=π k 2 Lβ A 2 ε 1/3 χ T ×( 6.6796{ Re [ Ν( R T , 11 6 ) + 1 ϖ 2 θ Ν( R S , 11 6 ) ( 1+θ ) ϖθ Ν ( R TS , 11 6 ) ] H( R S , 11 6 ) 1 ϖ 2 θ H( R T , 11 6 )+ ( 1+θ ) ϖθ H( R TS , 11 6 ) } +3.545 C 1 η 2/3 { Re [ Ν( R T , 3 2 ) + 1 ϖ 2 θ Ν( R S , 3 2 ) ( 1+θ ) ϖθ Ν( R TS , 3 2 ) ] H( R S , 3 2 ) 1 ϖ 2 θ H( R T , 3 2 )+ ( 1+θ ) ϖθ H( R TS , 3 2 ) } ),
H( R i ,α )= ( η R i ) 2α2 F 2 1 ( 1α, 1 2 ; 3 2 ; ΛL R i 2 k η 2 ).
Re[ F 2 1 ( 1α,1;2; iL R j 2 k η 2 + 2 3 ( iΛ+ Θ ˜ ) iL R j 2 k η 2 ) ]= 6Λ α( L R j 2 k η 2 )[ ( 1+2Θ ) 2 +4 Λ 2 ] + ( L R j 2 k η 2 ) 1 [ ( 1+2Θ ) 2 + ( 2Λ+3k η 2 /( L R j 2 ) ) 2 ] α/2 α 3 α1 [ ( 1+2Θ ) 2 +4 Λ 2 ] 1/2 sin( α φ 1 + φ 2 ),
σ I,l 2 ( L,η, L 0 )=π k 2 Lβ A 2 ε 1/3 χ T ×{ 6.6796 [ F( R T , 11 6 ) + 1 ϖ 2 θ F( R S , 11 6 ) ( 1+θ ) ϖθ F( R TS , 11 6 ) H( R S , 11 6 ) 1 ϖ 2 θ H( R T , 11 6 )+ ( 1+θ ) ϖθ H( R TS , 11 6 ) ] +3.545 C 1 η 2/3 [ F( R T , 3 2 ) + 1 ϖ 2 θ F( R S , 3 2 ) ( 1+θ ) ϖθ F( R TS , 3 2 ) H( R S , 3 2 ) 1 ϖ 2 θ H( R T , 3 2 )+ ( 1+θ ) ϖθ H( R TS , 3 2 ) ] },
F( R j ,α )= ( L R j 2 k η 2 ) 1 { [ ( 1+2Θ ) 2 + ( 2Λ+ 3k η 2 L R j 2 ) 2 ] α 2 α 3 α1 [ ( 1+2Θ ) 2 +4 Λ 2 ] 1/2 sin( α φ 1 + φ 2 ) 6Λ α[ ( 1+2Θ ) 2 +4 Λ 2 ] }
σ I,r 2 ( r,L,η, L 0 )=2πβ A 2 ε 1/3 χ T k 2 L 0 1 0 κ [ 1+ C 1 ( κη ) 2/3 ] ( κ 2 + κ L 0 2 ) 11/6 exp( ΛL κ 2 ξ 2 k ) ×[ I 0 ( 2Λrκξ )1 ]{ exp[ ( κη ) 2 R T 2 ]+ 1 ϖ 2 θ exp[ ( κη ) 2 R S 2 ] ( 1+θ ) ϖθ exp[ ( κη ) 2 R TS 2 ] }dκdξ.
I 0 ( x )=1+ n=1 ( x/2 ) 2n n!Γ( n+1 ) ,
σ I,r 2 ( r,L,η, L 0 )=2πβ A 2 ε 1/3 χ T k 2 L 0 1 n=1 ( Λrξ ) 2n n!Γ( n+1 ) 0 κ 2n+1 [ 1+ C 1 ( κη ) 2/3 ] ( κ 2 + κ L 0 2 ) 11/6 ×exp( ΛL κ 2 ξ 2 k ){ exp[ ( κη ) 2 R T 2 ]+ 1 ϖ 2 θ exp[ ( κη ) 2 R S 2 ] ( 1+θ ) ϖθ exp[ ( κη ) 2 R TS 2 ] }dκdξ.
σ I,r 2 ( r,L,η, L 0 )=2πβ A 2 ε 1/3 χ T k 2 L ×( 0 1 n=1 ( Λrξ ) 2n n! κ 0 2n5/3 { Γ( 5/6n ) Γ( 11/6 ) [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ]+ Γ( n5/6 ) Γ( n+1 ) κ 0 5/32n [ ( ΛL ξ 2 k + η 2 R T 2 ) 5/6n + 1 ϖ 2 θ ( ΛL ξ 2 k + η 2 R S 2 ) 5/6n ( 1+θ ) ϖθ ( ΛL ξ 2 k + η 2 R TS 2 ) 5/6n ] }dξ + C 1 η 2/3 0 1 n=1 ( Λrξ ) 2n Γ( n+4/3 ) n!Γ( n+1 ) κ 0 2n1 { Γ( 1/2n ) Γ( 11/6 ) [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ]+ Γ( n1/2 ) κ 0 12n Γ( n+4/3 ) × [ ( ΛL ξ 2 k + η 2 R T 2 ) 1/2n + 1 ϖ 2 θ ( ΛL ξ 2 k + η 2 R S 2 ) 1/2n ( 1+θ ) ϖθ ( ΛL ξ 2 k + η 2 R TS 2 ) 1/2n ] }dξ ).
0 1 ξ 2n ( ΛL ξ 2 k + η 2 R j 2 ) αn dξ = 1 2 ( η R j ) 2α2n Γ( n+1/2 ) Γ( n+3/2 ) F 2 1 ( nα,n+ 1 2 ;n+ 3 2 ; R j 2 ΛL η 2 k ),
σ I,r 2 ( r,L,η, L 0 )=πβ A 2 ε 1/3 χ T k 2 L ×( n=1 ( Λr ) 2n n! κ 0 2n5/3 { Γ( 5/6n ) ( 2n+1 )Γ( 11/6 ) [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ]+ κ 0 5/32n × Γ( n5/6 ) Γ( n+1 ) [ A( R T ,n, 5 6 ) + 1 ϖ 2 θ A( R S ,n, 5 6 ) ( 1+θ ) ϖθ A( R TS ,n, 5 6 ) ] } + C 1 η 2/3 n=1 ( Λr ) 2n Γ( n+4/3 ) n!Γ( n+1 ) κ 0 2n1 { Γ( 1/2n ) ( 2n+1 )Γ( 11/6 ) [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ] + Γ( n1/2 ) κ 0 12n Γ( n+4/3 ) [ A( R T ,n, 1 2 ) + 1 ϖ 2 θ A( R S ,n, 1 2 ) ( 1+θ ) ϖθ A( R TS ,n, 1 2 ) ] } ),
σ I,r 2 ( r,L,η, L 0 )2Λπβ A 2 ε 1/3 χ T k L 2 × r 2 w 2 ( { 2.4 κ 0 1/3 [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ] +5.5663 [ A( R T ,1, 5 6 ) + 1 ϖ 2 θ A( R S ,1, 5 6 ) ( 1+θ ) ϖθ A( R TS ,1, 5 6 ) ] }+1.1906 C 1 η 2/3 { 1.2562 κ 0 [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ] +1.4887 × [ A( R T ,1, 1 2 ) + 1 ϖ 2 θ A( R S ,1, 1 2 ) ( 1+θ ) ϖθ A( R TS ,1, 1 2 ) ] } ).
σ I,l 2 ( L,η, L 0 )π k 2 Lβ A 2 ε 1/3 χ T ×{ 6.6796 [ F( R T , 11 6 ) + 1 ϖ 2 θ F( R S , 11 6 ) ( 1+θ ) ϖθ F( R TS , 11 6 ) H( R S , 11 6 ) 1 ϖ 2 θ H( R T , 11 6 )+ ( 1+θ ) ϖθ H( R TS , 11 6 ) ] +3.545 C 1 η 2/3 [ F( R T , 3 2 ) + 1 ϖ 2 θ F( R S , 3 2 ) ( 1+θ ) ϖθ F( R TS , 3 2 ) H( R S , 3 2 ) 1 ϖ 2 θ H( R T , 3 2 )+ ( 1+θ ) ϖθ H( R TS , 3 2 ) ] }.
σ I,oc 2 ( r,L,η, L 0 )= σ I,r 2 ( r,L,η, L 0 )+ σ I,l 2 ( L,η, L 0 )2πβ A 2 ε 1/3 χ T k L 2 Λ ( r w ) 2 ×( { 2.4 κ 0 1/3 [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ] +5.5663 [ A( R T ,1, 5 6 ) + 1 ϖ 2 θ A( R S ,1, 5 6 ) ( 1+θ ) ϖθ × A( R TS ,1, 5 6 ) ] }+1.1906 Q 1 η 2/3 { 1.2562 κ 0 [ 1+ 1 ϖ 2 θ ( 1+θ ) ϖθ ] +1.4887 [ A( R T ,1, 1 2 ) + 1 ϖ 2 θ A( R S ,1, 1 2 ) ( 1+θ ) ϖθ A( R TS ,1, 1 2 ) ] } )+π k 2 Lβ A 2 ε 1/3 χ T { 6.6796 [ F( R T , 11 6 ) + 1 ϖ 2 θ F( R S , 11 6 ) ( 1+θ ) ϖθ F( R TS , 11 6 )H( R S , 11 6 ) 1 ϖ 2 θ H( R T , 11 6 )+ ( 1+θ ) ϖθ ×H( R TS , 11 6 ) ]+3.545 Q 1 η 2/3 [ F( R T , 3 2 ) + 1 ϖ 2 θ F( R S , 3 2 ) ( 1+θ ) ϖθ F( R TS , 3 2 ) H( R S , 3 2 ) 1 ϖ 2 θ H( R T , 3 2 )+ ( 1+θ ) ϖθ H( R TS , 3 2 ) ] }.

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