## Scattering by pure seawater at high salinity

Optics Express, Vol. 17, Issue 15, pp. 12685-12691 (2009)

http://dx.doi.org/10.1364/OE.17.012685

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### Abstract

A new model for seawater scattering was developed, in which Gibbs function was used exclusively to derive the thermodynamic parameters that are associated with density fluctuation. Because Gibbs function was determined empirically from highly accurate measurements of a group of thermodynamic variables and is valid for *S _{A}
* up to 120 g kg

^{−1}(Deep-Sea Research I, 55, 1639, 2008), we expect the model is also valid over the extended range of salinity. The model agrees with the measurements by Morel (Cahiers Oceanographiques, 20, 157, 1968) with an average difference of −0.6% for

*S*= 0 and 2.7% for

*S*= 38.4. The scattering by seawater as predicted increases with salinity in a non-linear fashion, and linear extrapolation of scattering based on Morel’s measurements would overestimate by up to 30%. The extrapolation of ZHH09 model (Optics Express, 17, 5698, 2009), which is valid for

*S*up to 40 g kg

_{A}^{−1}, however, agrees with the prediction within ± 2.5% over the entire range of salinity. Even though there are no measurements available for validation, the results suggested that the uncertainty is limited in using the newly developed model in estimating the scattering by seawater of high salinity.

© 2009 OSA

## 1. Introduction

*S*= 38.4 at five wavelengths of 366, 405, 436, 546, and 578 nm and found an average increase of 30% by seawater over pure water. For the past 4 decades, the measurements by Morel at one salinity value have been used throughout global oceans and coastal waters [e.g., 3], though sometimes adjusted linearly as a function of salinity [e.g., 4

4. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. **40**(30), 5503–5507 (2001). [CrossRef]

5. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express **17**(7), 5698–5710 (2009). [CrossRef] [PubMed]

6. A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. **338**(16), 1275–1298 (1910). [CrossRef]

*β*(90), is due to two independent contributions:

*β*(90) and

_{d}*β*(90) are the scattering due to density and concentration (mixing ratio) fluctuations, respectively. Following Zhang and Hu [7

_{c}7. X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express **17**(3), 1671–1678 (2009). [CrossRef] [PubMed]

5. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express **17**(7), 5698–5710 (2009). [CrossRef] [PubMed]

*β*(90) and

_{d}*β*(90) can be further expressed as,

_{c}*λ*is the wavelength,

*k*the Boltzmann constant,

*N*the Avogadro number,

_{A}*ρ*,

*n*,

*T*,

*β*,

_{T}*S*and

*f*(

*δ*) are the density, the refractive index in vacuum, the absolute temperature, the isothermal compressibility, the salinity and the Cabannes factor of seawater, respectively, and

*a*and

_{w}*M*are the activity and molecular weight of water in seawater, respectively.

_{w}^{−1}[8

8. B. W. Logan, and D. E. Cebulski, “Semimentary Environments of Shark Bay, Western Australian,” in *Carbon Sedimentation and Environments, Shark Bay, Western Australia*, B. W. Logan, G. R. Davies, J. F. Read, and D. E. Cebulski, eds. (The American Association of Petroleum Geologists, Tulsa, 1970), p. 205.

*ρ*,

*n*,

*β*, and

_{T}*a*) that have been empirically determined for

_{w}*T*up to 30 °C and

*S*up to 40. With no measurements available on scattering by seawater at high salinity, the uncertainty associated with extrapolation is unknown. Another minor issue is that Practical Salinity Scale

*S*used in Eq. (3) is undefined for

*S*> 42 [9

9. F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(1), 50–72 (2008). [CrossRef]

10. R. Feistel, “A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to 120 g kg^{−1},” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(12), 1639–1671 (2008). [CrossRef]

*g*(

*S*,

_{A}*T*,

_{c}*p*), where

*S*is the absolute salinity in g kg

_{A}^{−1},

*T*the temperature in Celsius, and

_{c}*p*sea pressure in Pa, has an extended range of validity for 0 ≤

*S*≤ 120 g kg

_{A}^{−1}, −12 °C ≤

*T*≤80 °C, and −0.1 Pa ≤

_{c}*p*≤ 100 MPa. Since the thermodynamic parameters,

*ρ*,

*β*, and

_{T}*a*as used in Eqs. (2) and (3) can all be derived mathematically from Gibbs energy through partial derivatives of

_{w}*g*with respect to

*S*,

_{A}*T*,

_{c}*p*, or their combinations, we expect that their values thus derived are also valid over the extended ranges. The purposes of this study are to develop a light scattering model for seawater that is based on Gibbs function and to estimate the scattering for high salinity waters.

## 2. A scattering model based on the Gibbs function

*m*, and the mass of salt

_{w}*m*, at temperature

_{s}*T*, and sea (gauge) pressure

_{c}*p*,

*µ*and

^{W}*µ*are the chemical potential of water in seawater and of salt in seawater, respectively, and

^{S}9. F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(1), 50–72 (2008). [CrossRef]

*µ*[11

^{W}11. R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. **58**(1), 43–114 (2003). [CrossRef]

*µ*[10

^{S}10. R. Feistel, “A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to 120 g kg^{−1},” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(12), 1639–1671 (2008). [CrossRef]

*g*(

^{W}*T*,

_{c}*p*) and

*g*(

^{S}*S*,

_{A}*T*,

_{c}*p*), Eq. (4) becomes

*ϕ*of seawater can be defined by the difference between the Gibbs potential of pure water and of water in seawater. Following Eq. (5).19) of Feistel and Marion [12

12. R. Feistel and G. M. Marion, “A Gibbs-Pitzer function for high-salinity seawater thermodynamics,” Prog. Oceanogr. **74**(4), 515–539 (2007). [CrossRef]

10. R. Feistel, “A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to 120 g kg^{−1},” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(12), 1639–1671 (2008). [CrossRef]

*m*is the molality (moles of salt per mass of water) and

*R*=

*N*×

_{A}*k*the molar gas constant. The water activity,

*a*, is linked with the osmotic pressure [13

_{w}13. F. J. Millero and W. H. Leung, “The thermodynamics of seawater at one atmosphere,” Am. J. Sci. **276**, 1035–1077 (1976). [CrossRef]

*s*,

*t*, or

*p*after the letter

*g*to denote the partial derivative of the Gibbs function with respect to the variables of

*S*,

_{A}*T*, or

_{c}*p*. Thermodynamically, the isothermal compressibility can be derived from the Gibbs energy as

*g*) and pressure (

_{ps}*g*) and the variation of chemical potential (

_{pp}*g*) with salinity (

_{s}*g*). The similarity between Eq. (2) and Eq. (11) simply reiterates the fact that the underlying physics for scattering by a pure liquid and by a mixture is the same: density fluctuation. With Eqs. (11) and (9), the total scattering by seawater, Eq. (1), can be rewritten as:

_{ss}*n*[7

7. X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express **17**(3), 1671–1678 (2009). [CrossRef] [PubMed]

*S*(0 – 40) and

*T*(0 – 30°C) under normal pressure [e.g., 14]. For a given temperature and pressure, the values of

_{c}*n*were found varying with salinity linearly [14,15

15. Y. Miyake, “Chemical studies of the Western Pacific Ocean. IV. The refractive index of sea water,” Bull. Chem. Soc. Jpn. **14**(6), 239–242 (1939). [CrossRef]

16. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. **34**(18), 3477–3480 (1995). [CrossRef] [PubMed]

*S*< 40 will be extrapolated to estimate

*n*over the extended salinity range. The details on how to estimate the terms

*n*can be found in Table 1 of Ref [5

5. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express **17**(7), 5698–5710 (2009). [CrossRef] [PubMed]

*g*is also empirically determined, it does ensure that the inter-relationships among different thermodynamic variables are always satisfied. The independent variables for Eq. (12) are temperature

*T*(or

_{c}*T*, T = 273.15 +

*T*), sea pressure

_{c}*p*(or absolute pressure

*P*,

*P*= 101325 Pa +

*p*), and absolute salinity

*S*(within the definition range of Practical Salinity Scale,

_{A}*S*,

*S*(g kg

_{A}^{−1}) = 1.0047

*S*[9

9. F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(1), 50–72 (2008). [CrossRef]

*β*(

*θ*), at the scattering angle,

*θ*, or the total scattering coefficient,

*b*, can be calculated as:

## 3. Results and discussion

*S*= 0 and

*S*= 38.4, at which the laboratory measurements were carried out by Morel [1,2]. The results are shown in Fig. 1 . A value of 0.039 for the depolarization ratio parameter,

*δ*[17

17. R. S. Farinato and R. L. Rowell, “New values of the light scattering depolarization and anisotropy of water,” J. Chem. Phys. **65**(2), 593–595 (1976). [CrossRef]

^{−1}calculated using Eq. (12) at 546 nm and 20°C is shown in Fig. 2 . For comparison, the estimates by ZHH09 model and by extrapolation of linear assumption [4, Eq. (3) in 5] are also shown. Overall, the scattering increases with salinity. But clearly the variation is non-linear, which is due to a combination of two factors: decreasing contribution due to density fluctuation and increasing contribution due to concentration fluctuation, with latter effect dominating [5

**17**(7), 5698–5710 (2009). [CrossRef] [PubMed]

*S*= 120 g kg

_{A}^{−1}. Twardowski (personal communication) observed a similar pattern of variation in scattering by NaCl solutions with concentrations up to 100 g kg

^{−1}. It is interesting to note that Eq. (12) and ZHH09 model agree to each other within ± 2.5% over the entire range of possible salinity values. Considering that the parameters used in the two models were derived quite differently, their close agreement with each other suggests that both models are theoretically sound and also confirms that the measurements that were used to derive the thermodynamic parameters were of high quality [10

^{−1},” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(12), 1639–1671 (2008). [CrossRef]

18. W. Wagner and A. Pruß, “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data **31**(2), 387–535 (2002). [CrossRef]

*S*up to 120 g kg

_{A}^{−1}, we recommend that 1) for

*S*(or

*S*) < 40 (or 40 g kg

_{A}^{−1}), use ZHH09 model; and for higher salinity, use Eq. (12).

^{−1},” Deep Sea Res. Part I Oceanogr. Res. Pap. **55**(12), 1639–1671 (2008). [CrossRef]

**55**(1), 50–72 (2008). [CrossRef]

*β*(Eq. (9)) and

_{T}*a*(Eq. (8)), and hence Eq. (12) associated with small chemical composition variations are also negligible. While we focused in this study on high salinity, Eq. (12) applies to seawater with high temperature or under high pressure as well. The variations of scattering with temperature or pressure, however, are much smaller as compared to the variations due to salinity. For example, the scattering decreases ~1.4% from surface to a depth of 1000 meters; and the scattering remains about the same (within 0.5%) between 26 °C and 60 °C. The spectral signature varies marginally with salinity (solely due to

_{w}*n*), with the power-law slope varying from −4.29 to −4.32 for

*S*from 0 to 120 g kg

_{A}^{−1}.

*δ*: its absolute value for pure water [7

7. X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express **17**(3), 1671–1678 (2009). [CrossRef] [PubMed]

**17**(7), 5698–5710 (2009). [CrossRef] [PubMed]

*δ*is between 0.03 and 0.05, roughly ± 25% variation about 0.039, a value adopted in this study,

*β*(90) or

*b*would vary approximately ± 2%. However, the exact behavior of

*δ*for seawater is still unknown. To further constrain the uncertainty in scattering by seawater, a fundamental quantity in ocean optics, the next challenge would be to improve our understanding of

*δ*and its variation with salinity.

## 4. Conclusions

^{−1}in absolute salinity. And we expect the model (Eq. (12)) based on Gibbs function would also be valid for the extended ranges. The purpose is to provide a first-order estimate of seawater scattering at high salinity because no measurements are available. The agreement of Eq. (12) with Morel’s measurements, −0.6% for

*S*= 0 and 2.7% for

*S*= 38.4 and consistent agreement with ZHH09 model for

*S*= 0 – 120 g kg

_{A}^{−1}, seemed to suggest the uncertainty in using Eq. (12) for high salinity waters is limited. The Matlab code implementing Eq. (12) can be downloaded at http://www.und.edu/instruct/zhang.

## Acknowledgements

## References and links

1. | A. Morel, “Etude Experimentale de la diffusion de la lumiere par l'eau, les solutions de chlorure de sodium et l'eau de mer optiquement pures,” J. Chim. Phys. |

2. | A. Morel, “Note au sujet des constantes de diffusion de la lumiere pour l'eau et l'eau de mer optiquement pures,” Cah. Oceanogr. |

3. | C. D. Mobley, |

4. | E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. |

5. | X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express |

6. | A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. |

7. | X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express |

8. | B. W. Logan, and D. E. Cebulski, “Semimentary Environments of Shark Bay, Western Australian,” in |

9. | F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep Sea Res. Part I Oceanogr. Res. Pap. |

10. | R. Feistel, “A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to 120 g kg |

11. | R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. |

12. | R. Feistel and G. M. Marion, “A Gibbs-Pitzer function for high-salinity seawater thermodynamics,” Prog. Oceanogr. |

13. | F. J. Millero and W. H. Leung, “The thermodynamics of seawater at one atmosphere,” Am. J. Sci. |

14. | R. W. Austin, and G. Halikas, “The index of refraction of seawater,” (Scripps Institute of Oceanography, La Jolla, 1974), p. 121. |

15. | Y. Miyake, “Chemical studies of the Western Pacific Ocean. IV. The refractive index of sea water,” Bull. Chem. Soc. Jpn. |

16. | X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. |

17. | R. S. Farinato and R. L. Rowell, “New values of the light scattering depolarization and anisotropy of water,” J. Chem. Phys. |

18. | W. Wagner and A. Pruß, “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.5840) Scattering : Scattering, molecules

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: June 2, 2009

Revised Manuscript: June 22, 2009

Manuscript Accepted: July 1, 2009

Published: July 20, 2009

**Citation**

Xiaodong Zhang and Lianbo Hu, "Scattering by pure seawater at high salinity," Opt. Express **17**, 12685-12691 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12685

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### References

- A. Morel, “Etude Experimentale de la diffusion de la lumiere par l'eau, les solutions de chlorure de sodium et l'eau de mer optiquement pures,” J. Chim. Phys. 10, 1359–1366 (1966).
- A. Morel, “Note au sujet des constantes de diffusion de la lumiere pour l'eau et l'eau de mer optiquement pures,” Cah. Oceanogr. 20, 157–162 (1968).
- C. D. Mobley, Light and water: radiative transfer in natural waters (Academic Press, San Diego, 1994).
- E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40(30), 5503–5507 (2001). [CrossRef]
- X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express 17(7), 5698–5710 (2009). [CrossRef] [PubMed]
- A. Einstein, “Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes,” Ann. Phys. 338(16), 1275–1298 (1910). [CrossRef]
- X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express 17(3), 1671–1678 (2009). [CrossRef] [PubMed]
- B. W. Logan and D. E. Cebulski, “Semimentary Environments of Shark Bay, Western Australian,” in Carbon Sedimentation and Environments, Shark Bay, Western Australia, B. W. Logan, G. R. Davies, J. F. Read, and D. E. Cebulski, eds. (The American Association of Petroleum Geologists, Tulsa, 1970), p. 205.
- F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale,” Deep Sea Res. Part I Oceanogr. Res. Pap. 55(1), 50–72 (2008). [CrossRef]
- R. Feistel, “A Gibbs function for seawater thermodynamics for −6 to 80 °C and salinity up to 120 g kg−1,” Deep Sea Res. Part I Oceanogr. Res. Pap. 55(12), 1639–1671 (2008). [CrossRef]
- R. Feistel, “A new extended Gibbs thermodynamic potential of seawater,” Prog. Oceanogr. 58(1), 43–114 (2003). [CrossRef]
- R. Feistel and G. M. Marion, “A Gibbs-Pitzer function for high-salinity seawater thermodynamics,” Prog. Oceanogr. 74(4), 515–539 (2007). [CrossRef]
- F. J. Millero and W. H. Leung, “The thermodynamics of seawater at one atmosphere,” Am. J. Sci. 276, 1035–1077 (1976). [CrossRef]
- R. W. Austin, and G. Halikas, “The index of refraction of seawater,” (Scripps Institute of Oceanography, La Jolla, 1974), p. 121.
- Y. Miyake, “Chemical studies of the Western Pacific Ocean. IV. The refractive index of sea water,” Bull. Chem. Soc. Jpn. 14(6), 239–242 (1939). [CrossRef]
- X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995). [CrossRef] [PubMed]
- R. S. Farinato and R. L. Rowell, “New values of the light scattering depolarization and anisotropy of water,” J. Chem. Phys. 65(2), 593–595 (1976). [CrossRef]
- W. Wagner and A. Pruß, “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data 31(2), 387–535 (2002). [CrossRef]

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