Abstract

The finite-element method has been applied to solving the radiative-transfer equation in a layered medium with a change in the refractive index, such as the atmosphere–ocean system. The physical processes that are included in the algorithm are multiple scattering, bottom-boundary bidirectional reflectivity, and refraction and reflection at the interface between the media with different refractive properties. The incident radiation is a parallel flux on the top boundary that is characteristic of illumination of the atmosphere by the Sun in the UV, visible, and near-IR regions of the electromagnetic spectrum. The necessary changes, compared with the case of a uniformly refracting layered medium, are described. An energy-conservation test has been performed on the model. The algorithm has also been validated through comparison with an equivalent backward Monte Carlo code and with data taken from the literature, and optimal agreement was shown. The results show that the model allows energy conservation independently of the adopted phase function, the number of grid points, and the relative refractive index. The radiative-transfer model can be applied to any other layered system with a change in the refractive index. The fortran code for this algorithm is documented and is available for applications.

© 1999 Optical Society of America

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References

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  1. V. Kisselev, L. Roberti, G. Perona, “Finite-element algorithm for radiative transfer in a vertically inhomogeneous medium: numerical scheme and applications,” Appl. Opt. 34, 8460–8471 (1995).
    [CrossRef] [PubMed]
  2. R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.
  3. O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).
  4. K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
    [CrossRef]
  5. K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2969–2706 (1981).
  6. K. Stamnes, S. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef] [PubMed]
  7. K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
    [CrossRef]
  8. B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).
  9. V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
    [CrossRef]
  10. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).
  11. C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
    [CrossRef]
  12. V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).
  13. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  14. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [CrossRef] [PubMed]
  15. L. Roberti, “Monte Carlo radiative transfer in the microwave and in the visible: biasing techniques,” Appl. Opt. 36, 7929–7938 (1997).
    [CrossRef]
  16. T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).
  17. O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.
  18. W. J. Wiscombe, “The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977).
    [CrossRef]
  19. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  20. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]

1997 (2)

1995 (1)

1994 (2)

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

1993 (1)

1988 (1)

1981 (2)

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2969–2706 (1981).

1977 (1)

W. J. Wiscombe, “The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1975 (1)

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973 (1)

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

Boynton, G. C.

Bulgarelli, B.

B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).

Dale, H.

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2969–2706 (1981).

Devaux, C.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

Fouquart, Y.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

Gentili, B.

Gordon, H. R.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Harringtion, R. F.

R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

Herman, M.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

Jayaweera, K.

Jin, Z.

Kattawar, G. W.

Kisselev, V.

Kisselev, V. B.

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Lenoble, J.

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

Liou, K. N.

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).

Mobley, C. D.

Morel, A.

Perona, G.

V. Kisselev, L. Roberti, G. Perona, “Finite-element algorithm for radiative transfer in a vertically inhomogeneous medium: numerical scheme and applications,” Appl. Opt. 34, 8460–8471 (1995).
[CrossRef] [PubMed]

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).

Reinersman, P.

Roberti, L.

Smotky, O. I.

O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.

Sobolev, V. V.

V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).

Stamnes, K.

Stavn, R. H.

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsay, S.

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

Appl. Opt. (6)

J. Atmos. Sci. (3)

W. J. Wiscombe, “The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2969–2706 (1981).

J. Geophys. Res. (1)

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

J. Quant. Spectrosc. Radiant. Transfer (1)

C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (7)

V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).

T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).

O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).

B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).

R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

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Figures (5)

Fig. 1
Fig. 1

Reflection and refraction at the air–water interface for upwelling [case (1)] and downwelling [case (2)] radiance (Subsection 2.D): nw, first water layer; η i , direction of incident radiation; η t direction of transmitted radiation. The + and - signs refer to the direction of propagation: +, downward radiation; -, upward radiation. The total internal reflection zone is indicated.

Fig. 2
Fig. 2

Radiance at the top of the atmosphere (W m-2 nm-1 sr-1) for the atmosphere–ocean system described in Subsection 3.C: -, FEM (N = 64); ○, Monte Carlo (106 photons).

Fig. 3
Fig. 3

Radiance (Wm-2 nm-1 sr-1) just above the air–water interface for the atmosphere–ocean system described in Subsection 3.C: -, FEM (N = 64); ○, Monte Carlo (106 photons).

Fig. 4
Fig. 4

Radiance (Wm-2 nm-1 sr-1) just below the air–water interface for the atmosphere–ocean system described in Subsection 3.C: -, FEM (N = 64); ○, Monte Carlo (106 photons).

Fig. 5
Fig. 5

Radiance (Wm-2 nm-1 sr-1) in the upper part of the water layer (τ = 1.6) for the atmosphere–ocean system described in Subsection 3.D: -, FEM (N = 64).

Tables (5)

Tables Icon

Table 1 Average Values of Ed, Eou, and Lu at Selected Depths Provided in Ref. 14 (Problem 3) with Their Sample Standard Deviation, and Corresponding Values Obtained with FEM (N = 4) and FEM (N = 64)

Tables Icon

Table 2 Total Flux (Wm-2nm-1) at Selected Depths Obtained with FEM (N = 2, N = 4, N = 8) for the Conservative System Described in Subsection 3.Ba

Tables Icon

Table 3 Mean absolute RE, max RE of the FEM Radiance Values for Different Numbers of Grid Points N with respect to the FEM (N = 64) Solution for the Atmosphere–Ocean Model of Subsection 3.D (∊ = 10-9)

Tables Icon

Table 4 CPU Times (Seconds) on an OpenVMS ALPHA V6.1 for Computation of One Harmonic for Different Values of N with Two Interpolation Angles, Four Layers, and Two Output Levels

Tables Icon

Table 5 Maximum Values of Each Harmonic for the Atmospheric–Ocean Model of Subsection 3.D at Different Output Levels as a Function of the Harmonic Number (∊ = 10-9)

Equations (67)

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Lτ, η, ϕ=m=0M Lmτ, ηcos mϕ.
Lm+τ, η=Lmτ, ηif η>0,Lm+τ, η=0if η<0,
Lm-τ, η=Lmτ, ηif η<0,Lm-τ, η=0if η>0.
pcos Θ=k=0M xkPkcos Θ=m=0Mk=mM xkmPkmηPkmηcos mϕ-ϕ=m=0M pmη, ηcos mϕ-ϕ,
cos Θ=ηη+1-η21/21-η21/2 cosϕ-ϕ,
xkm=2-δ0,mxkk-m!k+m!,
ηdLm±τ, η/dτ=-Lm±τ, η+Λl/2 -10 plm-η, ηLm-τ, ηdη+Λl/2 01 plm+η, ηLm+τ, ηdη+Qlmτ, η,
Qlmτ, ηatm=ΛlS/4plmη, ηoexp-τ/ηo+plmη, -ηoRCo exp-2τw-τ/ηo,
Qlmτ, ηocn=ΛlS/4plmη, ηtoTCo×exp-τw/ηoexp-τ-τw/ηto,
Lm-τ, η=i=1N Limτbimη,  Lm+τ, η=i=N+12N Limτbimη,
bimη=Pmmη1-|η-ηi|/hif |η-ηi|<h,0otherwise,
-11 PkmηPkmηdη=2/2k+1,
AmdLmτdτ=BmLmτ+Rmηoexp-τ/η0+RCoRm-ηoexp-2τw-τ/η0,
AmdLmτdτ=BmLmτ+TCoRmηto×exp-τw/ηoexp-τ-τw/ηto,
Llτ=Xlexp-τ/ηoWlηoXltRlηo+RCo×expτ-2τw/ηoWl-ηoXltRl-ηo+DlτN1l,
Llτ=XlTCo exp-τw/ηoexp-τ-τw/ηto×WlηtoXltRlηto+DlτN1l,
Dlτ=D1lτ00D1l-τ
D1lτ=diagexp-αilτ,  i=1, , N,
Wlx=W1lx00-W2lx
Wjlx=diag1/αil-1/x,  j=1, 2,
-αlAXl=BlXl.
Nl=D1lτl-100D1l-τl×N1l+E000WlηoXlRlηo+000Ediagexp2τwαj ×Wl-ηoXlRCoRl-ηo,
Nl=D1lτl-100D1l-τl×N1l+E000diagexpαjτw-τw/ηo×WlηtoXlTCoRlηto,
Llτ=XlD1lτ-τl-100D1lτl-τNl+Klτ, ηo00E exp-τ/ηo×E00-W2lηoXltRlηo+E expτ-2τw/ηo00Kl2τw-τ, ηo×-W2lηo00EXltRCoRl-ηo,
Llτ=XlD1lτ-τl-100D1lτl-τNl+Klτ-τw, ηto00E exp-τ-τw/ηto×E00-W2lηtoXltTCoRlηto ×exp-τw/ηo,
Klx, y=E exp-x/y-D1lxW1ly.
L1+0=0,
Lm-τ, η=2 01 rmη, ηLinm+τ, ηηdη+TCoSηtormη, ηto×exp-τw/ηo exp-τ-τw/ηto,
BˆL1-=SL2++RS,
Llτl=Ll+1τl.
Lnw-1-τw, ηt=TC1/nwa, ηi, ηtLnw-τw, ηi+RCnwa, ηt, ηiLnw-1+τw, ηt,
Lnw+τw, ηt=TCnwa, ηi, ηtLnw-1+τw, ηi+RC1/nwa, ηt, ηiLnw-τw, ηt,
RCx, ηt, ηi=12xηt-ηixηt+ηi2+ηt-xηiηt+xηi2,
TCx, ηi, ηt=2x3ηiηt1ηi+xηt2+1xηi+ηt2,
ηi=fx, ηt=1-x2+x2ηt21/2.
Tnw0-RnwD2Lnw-τwLnw+τw=D1-Rnw-10Tnw-1Lnw-1-τwLnw-1+τw,
Di,j1=01 bjηtbiηtdηt  i, j=1, , N,
Di,j2=01 bjηtbiηtdηt i, j=N+1, , 2N,
Ti,jnw-1=ηc1 bjηtTCfnwa, ηt, ηtbifnwa, ηtdηt,  i, j=N+1, , 2N,
Ti,jnw=01 bjηtTCf1/nwa, ηt, ηt×bif1/nwa, ηtdηt,  i, j=1, , N,
Ri,jnw=0ηc bjηtbiηtdηt+ηc1 bjηtRCfnwa, ηt, ηtbiηtdηt,  i=1, , N; j=N+1, , 2N,
Ri,jnw-1=01 bjηtRCf1/nwa, ηt, ηtbiηtdηt  i=N+1, , 2N; j=1, , N,
UN=V
N=N1N2···N,
η dLlτ, ηdτ=-Llτ, η+Zlτ, η,
Zlτ, η=Λl/2 -1+1 plη, ηLˆlτ, ηdη+Qlη, τ.
Llτ, η=exp-τ/ηη0τ expτ/ηZlτ, ηdτ,
Llτ, η=exp-τ/ηητwτ expτ/ηZlτ, ηdτ+TC1/nwa, ηref, ηLnw-1τw, ηref+RCnwa, η, ηrefLnwτw, -η×expτw-τ/η,
Zlτ, η=FlηLlτ+Qlτ, η,
Lτ, η=exp-τ/ηηPsτ, η+l=1s-1 Plτl, η,
Lτ, η=exp-τ/ηηPsτ, η+l=s-1nw Plτl, η+TCLnw-1τw, ηref+RCLnwτw, -η ×expτw-τ/η,
Ply, η=τl-1y expτ/ηZτ, ηdτ,
Ply, η=FlXlH1l00H2lNl+H3l0, ηo, αjl00EH4l0, ηo×E00-W2lηoXltRlηo+EH4l2τW, -ηo00-H3l2τW, -ηo, -αjl×-W2lηo00EXltRC0Rl-ηo+SΛl4H4l0, ηok=m xklPkmηPkmηo+RC0H4l2τW, -ηok=m xklPkmηPkm-ηo,
Ply, η=FlXlH1l00H2lNl+H3lτW, ηto, αjl00EH4lτW, ηto×E00-W2lηtoXltTC0Rlηtoexp-τW/ηo+SΛl4 TC0 exp-τW/ηo×H4lτW, ηtok=m xklPkmηPkmηto,
H1ly; αjl=expy/ηdiagexp-y-τl-1/η-exp-αjly-τl-1αjl-1/η,  H2ly; αjl=expy/ηdiagexp-αjlτl-y-exp-y-τl-1/η-αjlτl-τl-1αjl+1/η,  H3ly; v, x, αjl=EH4ly; v, x-diagH4ly; v, 1/αjdiag1/αjl-1/x,  H4ly; v, x=expy/ηexpv/x1/η-1/xexp-y/x- exp-y-τl-1/η-τl-1/x,
Llτ, η=-exp-τ/ηηττw expτ/ηZlτ, ηdτ+TCnwa, ηref, ηLnwτw, ηref+RC1/nwa, η, ηrefLnw-1τw, -η×expτw-τ/η,
Llτ, η=-exp-τ/ηηττL expτ/ηZlτ, ηdτ+L-τ, ηexpτ-τ/η,
H1ly; αjl=expy/ηdiagexpαjlτl-1-y-expαjlτl-1-τl+τl-y/ηαjl-1/η,  H2ly; αjl=expy/ηdiagexp-y-τl/η-expαjly-τlαjl+1/η,  H3ly; v, x, αjl=EH4ly; v, x-diagH4ly; v, 1/αjdiag1/αjl-1/x,  H4ly; v, x=expy/ηexpv/x1/η-1/xexpτl-y/η-τl/x- exp-τ/x,
Eouτ=Ψu Lτ, η, ϕdΩ.
Edτ=Ψd Lτ, η, ϕ|η|dΩ+πSηo exp-τ/ηofor ττw,TCoπSηto exp-τw/ηoexpτ-τw/ηtofor τ>τw,
Cz=Co+hs2π exp-12z-zmaxs2,
ap=0.04C0.602,  bp=0.33C0.620,
aw=0.0257,  bp=0.029.
pwcos Θ=341+cos2 Θ.
pcos Θ=bwbw+bpz pwcos Θ+bpzbw+bpz pPcos Θ,
λ=bw+bpzaw+apz+bw+bw+bpz.
RE=FEMN=iFEMN=64-1,  i=4, 8, 16, 32.

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