Abstract

The scattering and backscattering properties of bubble populations in the upper ocean are estimated with Mie theory and a generalized bubble size spectrum based on in situ observations. Optical properties of both clean bubbles and bubbles coated with an organic film are analyzed; the results are compared with the corresponding optical properties of micro-organisms of similar size. Given a bubble number density (from ∼105 to ∼107 m-3) frequently found at sea, the bubble populations significantly influence the scattering process in the ocean, especially in oligotrophic waters. Bubbles appear to make a large contribution to the missing terms in constructing the observed total backscattering coefficient of the ocean. This contribution to backscattering is strongly enhanced if the bubbles are coated with organic film. The injection of bubbles will shift ocean color toward the green, resembling phytoplankton blooms, and hence introducing error in ocean color remote sensing if its effect is not corrected.

© 1998 Optical Society of America

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    [CrossRef]
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  54. B. D. Johnson, P. J. Wangersky, “Microbubbles: stabilization by monolayers of adsorbed particles,” J. Geophys. Res. 92, 14,641–14,647 (1987).
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  55. F. E. Fox, K. Herzfeld, “Gas bubbles with organic skin as cavitation nuclei,” J. Acoust. Soc. Am. 26, 984–989 (1954).
    [CrossRef]
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  61. According to Mie theory, the optical efficiency depends only on refractive index and a size factor x = 2πr/λ, where r is the particle radius and λ is the wavelength. They have an equivalent but reverse effect on x. For example, that the optical efficiency is constant in the visible (400–700 nm) at r = 50 μm only requires that it does not change for r from 39 to 69 μm at λ = 550 nm. For backscattering coefficient of bubbles, this requirement is almost always satisfied for both clean and dirty bubbles (Fig. 4).
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1996 (1)

1995 (2)

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

J. S. Cleveland, “Regional models for phytoplankton absorption as a function of chlorophyll a concentration,” J. Geophys. Res. 100, 13,333–13,344 (1995).
[CrossRef]

1994 (1)

1993 (1)

D. M. Farmer, C. L. McNeil, B. D. Johnson, “Evidence for the importance of bubbles in increasing air-sea gas flux,” Nature (London) 361, 620–623 (1993).
[CrossRef]

1992 (3)

Y. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

J. Wu, “Bubble flux and marine aerosol spectra under various wind velocities,” J. Geophys. Res. 97, 2327–2333 (1992).
[CrossRef]

S. A. Thorpe, P. Bowyer, D. K. Woolf, “Some factors affecting the size distributions of oceanic bubbles,” J. Phys. Oceanogr. 22, 382–389 (1992).
[CrossRef]

1991 (5)

A. Morel, Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

T. Platt, C. Caverhill, S. Sathyendranath, “Basin-scale estimates of oceanic primary production by remote sensing: the North Atlantic.” J. Geophys. Res. 96, 15,147–15,160 (1991).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

W. Arnott, P. L. Marston, “Unfolded optical glory of spheroids: backscattering of laser light from freely rising spheroidal air bubbles in water,” Appl. Opt. 30, 3429–3442 (1991).
[CrossRef] [PubMed]

1990 (1)

A. Morel, Y. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

1989 (2)

H. Medwin, N. D. Breitz, “Ambient and transient bubble spectral densities in quiescent seas and under spilling breaker,” J. Geophys. Res. 94, 12,751–12,759 (1989).
[CrossRef]

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

1988 (6)

T. J. O’Hern, L. d’Agostino, A. J. Acosta, “Comparison of holographic and coulter counter measurement of cavitation nuclei in the ocean,” Trans. ASME J. Fluids Eng. 110, 200–207 (1988).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

A. Morel, “Optical modeling of upper ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 48, 145–175 (1988).

D. C. Blanchard, L. D. Syzdek, “Film drop production as a function of bubble size,” J. Geophys. Res. 93, 3649–3654 (1988).
[CrossRef]

J. Wu, “Bubbles in the near-surface ocean: a general description,” J. Geophys. Res. 93, 587–590 (1988).
[CrossRef]

W. Arnott, P. L. Marston, “Optical glory of small freely rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
[CrossRef]

1987 (3)

B. D. Johnson, P. J. Wangersky, “Microbubbles: stabilization by monolayers of adsorbed particles,” J. Geophys. Res. 92, 14,641–14,647 (1987).
[CrossRef]

G. B. Crawford, D. M. Farmer, “On the spatial distribution of ocean bubbles,” J. Geophys. Res. 92, 8231–8243 (1987).
[CrossRef]

A. L. Walsh, P. J. Mulhearn, “Photographic measurements of bubble populations from breaking wind waves at sea,” J. Geophys. Res. 92, 14,553–14,565 (1987).
[CrossRef]

1984 (1)

D. E. Yount, E. W. Gillary, D. C. Hoffman, “A microscopic investigation of bubble formation nuclei,” J. Acoust. Soc. Am. 76, 1511–1521 (1984).
[CrossRef]

1983 (1)

R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. 74, 980–986 (1983).
[CrossRef]

1982 (2)

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[CrossRef]

S. A. Thorpe, “On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air-sea gas transfer,” Philos. Trans. R. Soc. London Ser. A 304, 155–210 (1982).
[CrossRef]

1981 (5)

J. Wu, “Bubble populations and spectra in near-surface ocean: summary and review of field measurements,” J. Geophys. Res. 86, 457–463 (1981).
[CrossRef]

P. L. Marston, D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981).
[CrossRef]

D. L. Kingsbury, P. L. Marston, “Mie scattering near the critical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
[CrossRef]

B. D. Johnson, R. C. Cooke, “Generation of stabilized microbubbles in seawater,” Science 213, 209–211 (1981).
[CrossRef] [PubMed]

P. J. Mulhearn, “Distribution of microbubbles in coastal waters,” J. Geophys. Res. 86, 6429–6434 (1981).
[CrossRef]

1979 (3)

R. A. Meyer, “Light scattering from biological cells: dependence of backscattering radiation on membrane thickness and refractive index,” Appl. Opt. 18, 585–588 (1979).
[CrossRef] [PubMed]

D. E. Yount, “Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei,” J. Acoust. Soc. Am. 65, 1429–1439 (1979).
[CrossRef]

B. D. Johnson, R. C. Cooke, “Bubble populations and spectra in coastal waters: a photographic approach,” J. Geophys. Res. 84, 3761–3766 (1979).
[CrossRef]

1977 (2)

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82, 971–976 (1977).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

1976 (1)

D. A. Kolovayev, “Investigation of the concentration and statistical size distribution of wind-produced bubbles in the near-surface ocean,” Oceanology 15, 659–661 (1976).

1975 (1)

1974 (1)

1973 (1)

O. B. Brown, H. R. Gordon, “Two component Mie scattering models of Sargasso Sea particles,” Limnol. Oceanogr. 17, 826–832 (1973).

1970 (1)

H. Medwin, “In situ acoustic measurements of bubble populations in coastal ocean waters,” J. Geophys. Res. 75, 599–611 (1970).
[CrossRef]

1957 (1)

D. C. Blanchard, A. H. Woodcock, “Bubble formation and modification in the sea and its meteorological significance,” Tellus 9, 145–158 (1957).
[CrossRef]

1954 (1)

F. E. Fox, K. Herzfeld, “Gas bubbles with organic skin as cavitation nuclei,” J. Acoust. Soc. Am. 26, 984–989 (1954).
[CrossRef]

Acosta, A. J.

T. J. O’Hern, L. d’Agostino, A. J. Acosta, “Comparison of holographic and coulter counter measurement of cavitation nuclei in the ocean,” Trans. ASME J. Fluids Eng. 110, 200–207 (1988).
[CrossRef]

Ahn, Y.

Y. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

A. Morel, Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

A. Morel, Y. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

Arnott, W.

Babin, M.

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

Billette, S.

P. L. Marston, S. Billette, C. Dean, “Scattering of light by a coated bubble in water near the critical and Brewster scattering angles,” in Ocean Optics IX, M. A. Blizard, ed., Proc. SPIE925, 308–316 (1988).
[CrossRef]

Blanchard, D. C.

D. C. Blanchard, L. D. Syzdek, “Film drop production as a function of bubble size,” J. Geophys. Res. 93, 3649–3654 (1988).
[CrossRef]

D. C. Blanchard, A. H. Woodcock, “Bubble formation and modification in the sea and its meteorological significance,” Tellus 9, 145–158 (1957).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bowyer, P.

S. A. Thorpe, P. Bowyer, D. K. Woolf, “Some factors affecting the size distributions of oceanic bubbles,” J. Phys. Oceanogr. 22, 382–389 (1992).
[CrossRef]

Breitz, N. D.

H. Medwin, N. D. Breitz, “Ambient and transient bubble spectral densities in quiescent seas and under spilling breaker,” J. Geophys. Res. 94, 12,751–12,759 (1989).
[CrossRef]

Bricaud, A.

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

Y. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing applications,” in Oceanography from Space, J. F. R. Gower, ed. (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

O. B. Brown, H. R. Gordon, “Size-refractive index distribution of clear coastal water particulates from light scattering,” Appl. Opt. 13, 2874–2881 (1974).
[CrossRef] [PubMed]

O. B. Brown, H. R. Gordon, “Two component Mie scattering models of Sargasso Sea particles,” Limnol. Oceanogr. 17, 826–832 (1973).

Caverhill, C.

T. Platt, C. Caverhill, S. Sathyendranath, “Basin-scale estimates of oceanic primary production by remote sensing: the North Atlantic.” J. Geophys. Res. 96, 15,147–15,160 (1991).
[CrossRef]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

Claustre, H.

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

Cleveland, J. S.

J. S. Cleveland, “Regional models for phytoplankton absorption as a function of chlorophyll a concentration,” J. Geophys. Res. 100, 13,333–13,344 (1995).
[CrossRef]

Cohen, L. H.

G. De Leeuw, L. H. Cohen, “Bubble size distribution in coastal seas,” in Air-Water Gas Transfer, B. Jahne, E. C. Monahan, eds. (AEON Verlag and Studio, Hanau, Germany, 1995) pp. 325–336.

Cooke, R. C.

B. D. Johnson, R. C. Cooke, “Generation of stabilized microbubbles in seawater,” Science 213, 209–211 (1981).
[CrossRef] [PubMed]

B. D. Johnson, R. C. Cooke, “Bubble populations and spectra in coastal waters: a photographic approach,” J. Geophys. Res. 84, 3761–3766 (1979).
[CrossRef]

Crawford, G. B.

G. B. Crawford, D. M. Farmer, “On the spatial distribution of ocean bubbles,” J. Geophys. Res. 92, 8231–8243 (1987).
[CrossRef]

d’Agostino, L.

T. J. O’Hern, L. d’Agostino, A. J. Acosta, “Comparison of holographic and coulter counter measurement of cavitation nuclei in the ocean,” Trans. ASME J. Fluids Eng. 110, 200–207 (1988).
[CrossRef]

De Leeuw, G.

G. De Leeuw, L. H. Cohen, “Bubble size distribution in coastal seas,” in Air-Water Gas Transfer, B. Jahne, E. C. Monahan, eds. (AEON Verlag and Studio, Hanau, Germany, 1995) pp. 325–336.

Dean, C.

P. L. Marston, S. Billette, C. Dean, “Scattering of light by a coated bubble in water near the critical and Brewster scattering angles,” in Ocean Optics IX, M. A. Blizard, ed., Proc. SPIE925, 308–316 (1988).
[CrossRef]

Dugdale, R. C.

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

Esaias, W. E.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

Farmer, D. M.

D. M. Farmer, C. L. McNeil, B. D. Johnson, “Evidence for the importance of bubbles in increasing air-sea gas flux,” Nature (London) 361, 620–623 (1993).
[CrossRef]

G. B. Crawford, D. M. Farmer, “On the spatial distribution of ocean bubbles,” J. Geophys. Res. 92, 8231–8243 (1987).
[CrossRef]

Feldman, G. C.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

Fox, F. E.

F. E. Fox, K. Herzfeld, “Gas bubbles with organic skin as cavitation nuclei,” J. Acoust. Soc. Am. 26, 984–989 (1954).
[CrossRef]

Gentili, B.

Gillary, E. W.

D. E. Yount, E. W. Gillary, D. C. Hoffman, “A microscopic investigation of bubble formation nuclei,” J. Acoust. Soc. Am. 76, 1511–1521 (1984).
[CrossRef]

Glazman, R. E.

R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. 74, 980–986 (1983).
[CrossRef]

Gordon, H. R.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

O. B. Brown, H. R. Gordon, “Size-refractive index distribution of clear coastal water particulates from light scattering,” Appl. Opt. 13, 2874–2881 (1974).
[CrossRef] [PubMed]

O. B. Brown, H. R. Gordon, “Two component Mie scattering models of Sargasso Sea particles,” Limnol. Oceanogr. 17, 826–832 (1973).

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, a Review (Springer-Verlag, New York, 1983).
[CrossRef]

Gregg, W. W.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

Herzfeld, K.

F. E. Fox, K. Herzfeld, “Gas bubbles with organic skin as cavitation nuclei,” J. Acoust. Soc. Am. 26, 984–989 (1954).
[CrossRef]

Hoffman, D. C.

D. E. Yount, E. W. Gillary, D. C. Hoffman, “A microscopic investigation of bubble formation nuclei,” J. Acoust. Soc. Am. 76, 1511–1521 (1984).
[CrossRef]

Hooker, S. B.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jacobs, M. M.

Johnson, B. D.

D. M. Farmer, C. L. McNeil, B. D. Johnson, “Evidence for the importance of bubbles in increasing air-sea gas flux,” Nature (London) 361, 620–623 (1993).
[CrossRef]

B. D. Johnson, P. J. Wangersky, “Microbubbles: stabilization by monolayers of adsorbed particles,” J. Geophys. Res. 92, 14,641–14,647 (1987).
[CrossRef]

B. D. Johnson, R. C. Cooke, “Generation of stabilized microbubbles in seawater,” Science 213, 209–211 (1981).
[CrossRef] [PubMed]

B. D. Johnson, R. C. Cooke, “Bubble populations and spectra in coastal waters: a photographic approach,” J. Geophys. Res. 84, 3761–3766 (1979).
[CrossRef]

B. D. Johnson, “Bubble populations: background and breaking waves,” in Oceanic Whitecaps and Their Role in Air-Sea Exchange Processes, E. C. Monahan, G. Mac Niocaill, eds. (Reidel, Dordrecht, The Netherlands, 1986), pp. 69–73.

Kiefer, D. A.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Kingsbury, D. L.

Kolovayev, D. A.

D. A. Kolovayev, “Investigation of the concentration and statistical size distribution of wind-produced bubbles in the near-surface ocean,” Oceanology 15, 659–661 (1976).

Langley, D. S.

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[CrossRef]

Lewis, M. R.

M. R. Lewis, “Satellite ocean color observations of global biogeochemical cycles,” in Primary Productivity and Biogeochemical Cycles in the Sea, P. G. Falkowski, A. Woodhead, eds. (Plenum, New York, 1992), pp. 139–154.

MacIntyre, F.

F. MacIntyre, “On reconciling optical and acoustical bubble spectra in the mixed layer,” in Oceanic Whitecaps and Their Role in Air-Sea Exchange Processes, E. C. Monahan, G. Mac Niocaill, eds. (Reidel, Dordrecht, The Netherlands, 1986), pp. 75–94.

Marston, P. L.

McClain, C. R.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

McNeil, C. L.

D. M. Farmer, C. L. McNeil, B. D. Johnson, “Evidence for the importance of bubbles in increasing air-sea gas flux,” Nature (London) 361, 620–623 (1993).
[CrossRef]

Medwin, H.

H. Medwin, N. D. Breitz, “Ambient and transient bubble spectral densities in quiescent seas and under spilling breaker,” J. Geophys. Res. 94, 12,751–12,759 (1989).
[CrossRef]

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82, 971–976 (1977).
[CrossRef]

H. Medwin, “In situ acoustic measurements of bubble populations in coastal ocean waters,” J. Geophys. Res. 75, 599–611 (1970).
[CrossRef]

Meyer, R. A.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

Monahan, E. C.

E. C. Monahan, “Near surface bubble concentration and oceanic whitecap coverage,” in Seventh Conference on Ocean-Atmosphere Interaction (American Meteorological Society, Boston, Mass., 1988), pp. 178–181.

E. C. Monahan, “The ocean as a source for atmospheric particles,” in The Role of Air-Sea Exchange in Geochemical Cycling, P. Buat-Menard, ed. (Reidel, Dordrecht, The Netherlands, 1986), pp. 129–163.
[CrossRef]

Morel, A.

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[CrossRef] [PubMed]

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

Y. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

A. Morel, Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

A. Morel, Y. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

A. Morel, “Optical modeling of upper ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 48, 145–175 (1988).

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, a Review (Springer-Verlag, New York, 1983).
[CrossRef]

A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing applications,” in Oceanography from Space, J. F. R. Gower, ed. (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

A. Morel, “Optics of marine particles and marine optics,” in Particle Analysis in Oceanography, S. Demers, ed. (Springer-Verlag, Berlin, 1990), pp. 141–188.

Mulhearn, P. J.

A. L. Walsh, P. J. Mulhearn, “Photographic measurements of bubble populations from breaking wind waves at sea,” J. Geophys. Res. 92, 14,553–14,565 (1987).
[CrossRef]

P. J. Mulhearn, “Distribution of microbubbles in coastal waters,” J. Geophys. Res. 86, 6429–6434 (1981).
[CrossRef]

O’Hern, T. J.

T. J. O’Hern, L. d’Agostino, A. J. Acosta, “Comparison of holographic and coulter counter measurement of cavitation nuclei in the ocean,” Trans. ASME J. Fluids Eng. 110, 200–207 (1988).
[CrossRef]

Platt, T.

O. Ulloa, S. Sathyendranath, T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
[CrossRef] [PubMed]

T. Platt, C. Caverhill, S. Sathyendranath, “Basin-scale estimates of oceanic primary production by remote sensing: the North Atlantic.” J. Geophys. Res. 96, 15,147–15,160 (1991).
[CrossRef]

Prieur, L.

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

Sathyendranath, S.

O. Ulloa, S. Sathyendranath, T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
[CrossRef] [PubMed]

T. Platt, C. Caverhill, S. Sathyendranath, “Basin-scale estimates of oceanic primary production by remote sensing: the North Atlantic.” J. Geophys. Res. 96, 15,147–15,160 (1991).
[CrossRef]

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

Stramski, D.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

D. Stramski, “Gas microbubbles: an assessment of their significance to light scattering in quiescent seas,” in Ocean Optics XII, J. S. Jaffe, eds., Proc. SPIE2258, 704–710 (1994).
[CrossRef]

Syzdek, L. D.

D. C. Blanchard, L. D. Syzdek, “Film drop production as a function of bubble size,” J. Geophys. Res. 93, 3649–3654 (1988).
[CrossRef]

Thorpe, S. A.

S. A. Thorpe, P. Bowyer, D. K. Woolf, “Some factors affecting the size distributions of oceanic bubbles,” J. Phys. Oceanogr. 22, 382–389 (1992).
[CrossRef]

S. A. Thorpe, “On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air-sea gas transfer,” Philos. Trans. R. Soc. London Ser. A 304, 155–210 (1982).
[CrossRef]

Ulloa, O.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Walsh, A. L.

A. L. Walsh, P. J. Mulhearn, “Photographic measurements of bubble populations from breaking wind waves at sea,” J. Geophys. Res. 92, 14,553–14,565 (1987).
[CrossRef]

Wangersky, P. J.

B. D. Johnson, P. J. Wangersky, “Microbubbles: stabilization by monolayers of adsorbed particles,” J. Geophys. Res. 92, 14,641–14,647 (1987).
[CrossRef]

Wilkerson, F. P.

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

Woodcock, A. H.

D. C. Blanchard, A. H. Woodcock, “Bubble formation and modification in the sea and its meteorological significance,” Tellus 9, 145–158 (1957).
[CrossRef]

Woolf, D. K.

S. A. Thorpe, P. Bowyer, D. K. Woolf, “Some factors affecting the size distributions of oceanic bubbles,” J. Phys. Oceanogr. 22, 382–389 (1992).
[CrossRef]

Wu, J.

J. Wu, “Bubble flux and marine aerosol spectra under various wind velocities,” J. Geophys. Res. 97, 2327–2333 (1992).
[CrossRef]

J. Wu, “Bubbles in the near-surface ocean: a general description,” J. Geophys. Res. 93, 587–590 (1988).
[CrossRef]

J. Wu, “Bubble populations and spectra in near-surface ocean: summary and review of field measurements,” J. Geophys. Res. 86, 457–463 (1981).
[CrossRef]

Yount, D. E.

D. E. Yount, E. W. Gillary, D. C. Hoffman, “A microscopic investigation of bubble formation nuclei,” J. Acoust. Soc. Am. 76, 1511–1521 (1984).
[CrossRef]

D. E. Yount, “Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei,” J. Acoust. Soc. Am. 65, 1429–1439 (1979).
[CrossRef]

Appl. Opt. (7)

Appl. Sci. Res. (1)

P. L. Marston, D. S. Langley, D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[CrossRef]

Deep-Sea Res. (1)

Y. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

J. Acoust. Soc. Am. (4)

R. E. Glazman, “Effects of adsorbed films on gas bubble radial oscillations,” J. Acoust. Soc. Am. 74, 980–986 (1983).
[CrossRef]

D. E. Yount, E. W. Gillary, D. C. Hoffman, “A microscopic investigation of bubble formation nuclei,” J. Acoust. Soc. Am. 76, 1511–1521 (1984).
[CrossRef]

D. E. Yount, “Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei,” J. Acoust. Soc. Am. 65, 1429–1439 (1979).
[CrossRef]

F. E. Fox, K. Herzfeld, “Gas bubbles with organic skin as cavitation nuclei,” J. Acoust. Soc. Am. 26, 984–989 (1954).
[CrossRef]

J. Geophys. Res. (18)

B. D. Johnson, R. C. Cooke, “Bubble populations and spectra in coastal waters: a photographic approach,” J. Geophys. Res. 84, 3761–3766 (1979).
[CrossRef]

H. Medwin, N. D. Breitz, “Ambient and transient bubble spectral densities in quiescent seas and under spilling breaker,” J. Geophys. Res. 94, 12,751–12,759 (1989).
[CrossRef]

G. B. Crawford, D. M. Farmer, “On the spatial distribution of ocean bubbles,” J. Geophys. Res. 92, 8231–8243 (1987).
[CrossRef]

P. J. Mulhearn, “Distribution of microbubbles in coastal waters,” J. Geophys. Res. 86, 6429–6434 (1981).
[CrossRef]

B. D. Johnson, P. J. Wangersky, “Microbubbles: stabilization by monolayers of adsorbed particles,” J. Geophys. Res. 92, 14,641–14,647 (1987).
[CrossRef]

J. Wu, “Bubble flux and marine aerosol spectra under various wind velocities,” J. Geophys. Res. 97, 2327–2333 (1992).
[CrossRef]

J. Wu, “Bubbles in the near-surface ocean: a general description,” J. Geophys. Res. 93, 587–590 (1988).
[CrossRef]

A. L. Walsh, P. J. Mulhearn, “Photographic measurements of bubble populations from breaking wind waves at sea,” J. Geophys. Res. 92, 14,553–14,565 (1987).
[CrossRef]

R. C. Dugdale, A. Morel, A. Bricaud, F. P. Wilkerson, “Modeling new production in upwelling centers: a case study of modeling new production from remotely sensed temperature and color,” J. Geophys. Res. 94, 18,119–18,132 (1989).
[CrossRef]

T. Platt, C. Caverhill, S. Sathyendranath, “Basin-scale estimates of oceanic primary production by remote sensing: the North Atlantic.” J. Geophys. Res. 96, 15,147–15,160 (1991).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytical radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).

A. Morel, “Optical modeling of upper ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 48, 145–175 (1988).

A. Bricaud, M. Babin, A. Morel, H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13,321–13,332 (1995).
[CrossRef]

J. S. Cleveland, “Regional models for phytoplankton absorption as a function of chlorophyll a concentration,” J. Geophys. Res. 100, 13,333–13,344 (1995).
[CrossRef]

H. Medwin, “In situ acoustic measurements of bubble populations in coastal ocean waters,” J. Geophys. Res. 75, 599–611 (1970).
[CrossRef]

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82, 971–976 (1977).
[CrossRef]

D. C. Blanchard, L. D. Syzdek, “Film drop production as a function of bubble size,” J. Geophys. Res. 93, 3649–3654 (1988).
[CrossRef]

J. Wu, “Bubble populations and spectra in near-surface ocean: summary and review of field measurements,” J. Geophys. Res. 86, 457–463 (1981).
[CrossRef]

J. Mar. Res. (2)

A. Morel, Y. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

A. Morel, Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Phys. Oceanogr. (1)

S. A. Thorpe, P. Bowyer, D. K. Woolf, “Some factors affecting the size distributions of oceanic bubbles,” J. Phys. Oceanogr. 22, 382–389 (1992).
[CrossRef]

Limnol. Oceanogr. (2)

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

O. B. Brown, H. R. Gordon, “Two component Mie scattering models of Sargasso Sea particles,” Limnol. Oceanogr. 17, 826–832 (1973).

Nature (London) (1)

D. M. Farmer, C. L. McNeil, B. D. Johnson, “Evidence for the importance of bubbles in increasing air-sea gas flux,” Nature (London) 361, 620–623 (1993).
[CrossRef]

Oceanology (1)

D. A. Kolovayev, “Investigation of the concentration and statistical size distribution of wind-produced bubbles in the near-surface ocean,” Oceanology 15, 659–661 (1976).

Philos. Trans. R. Soc. London Ser. A (1)

S. A. Thorpe, “On the clouds of bubbles formed by breaking wind-waves in deep water, and their role in air-sea gas transfer,” Philos. Trans. R. Soc. London Ser. A 304, 155–210 (1982).
[CrossRef]

Prog. Oceanogr. (1)

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Science (1)

B. D. Johnson, R. C. Cooke, “Generation of stabilized microbubbles in seawater,” Science 213, 209–211 (1981).
[CrossRef] [PubMed]

Tellus (1)

D. C. Blanchard, A. H. Woodcock, “Bubble formation and modification in the sea and its meteorological significance,” Tellus 9, 145–158 (1957).
[CrossRef]

Trans. ASME J. Fluids Eng. (1)

T. J. O’Hern, L. d’Agostino, A. J. Acosta, “Comparison of holographic and coulter counter measurement of cavitation nuclei in the ocean,” Trans. ASME J. Fluids Eng. 110, 200–207 (1988).
[CrossRef]

Other (16)

E. C. Monahan, “Near surface bubble concentration and oceanic whitecap coverage,” in Seventh Conference on Ocean-Atmosphere Interaction (American Meteorological Society, Boston, Mass., 1988), pp. 178–181.

B. D. Johnson, “Bubble populations: background and breaking waves,” in Oceanic Whitecaps and Their Role in Air-Sea Exchange Processes, E. C. Monahan, G. Mac Niocaill, eds. (Reidel, Dordrecht, The Netherlands, 1986), pp. 69–73.

F. MacIntyre, “On reconciling optical and acoustical bubble spectra in the mixed layer,” in Oceanic Whitecaps and Their Role in Air-Sea Exchange Processes, E. C. Monahan, G. Mac Niocaill, eds. (Reidel, Dordrecht, The Netherlands, 1986), pp. 75–94.

M. R. Lewis, “Satellite ocean color observations of global biogeochemical cycles,” in Primary Productivity and Biogeochemical Cycles in the Sea, P. G. Falkowski, A. Woodhead, eds. (Plenum, New York, 1992), pp. 139–154.

A. Morel, A. Bricaud, “Theoretical results concerning the optics of phytoplankton, with special reference to remote sensing applications,” in Oceanography from Space, J. F. R. Gower, ed. (Plenum, New York, 1981), pp. 313–327.
[CrossRef]

D. Stramski, “Gas microbubbles: an assessment of their significance to light scattering in quiescent seas,” in Ocean Optics XII, J. S. Jaffe, eds., Proc. SPIE2258, 704–710 (1994).
[CrossRef]

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, a Review (Springer-Verlag, New York, 1983).
[CrossRef]

G. De Leeuw, L. H. Cohen, “Bubble size distribution in coastal seas,” in Air-Water Gas Transfer, B. Jahne, E. C. Monahan, eds. (AEON Verlag and Studio, Hanau, Germany, 1995) pp. 325–336.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

According to Mie theory, the optical efficiency depends only on refractive index and a size factor x = 2πr/λ, where r is the particle radius and λ is the wavelength. They have an equivalent but reverse effect on x. For example, that the optical efficiency is constant in the visible (400–700 nm) at r = 50 μm only requires that it does not change for r from 39 to 69 μm at λ = 550 nm. For backscattering coefficient of bubbles, this requirement is almost always satisfied for both clean and dirty bubbles (Fig. 4).

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, “An overview of SeaWiFS and ocean color,” in SeaWiFS Technical Report Series, Vol. 1, NASA Tech. Memo. 104566 (NASA, Greenbelt, Md., 1992).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

P. L. Marston, S. Billette, C. Dean, “Scattering of light by a coated bubble in water near the critical and Brewster scattering angles,” in Ocean Optics IX, M. A. Blizard, ed., Proc. SPIE925, 308–316 (1988).
[CrossRef]

A. Morel, “Optics of marine particles and marine optics,” in Particle Analysis in Oceanography, S. Demers, ed. (Springer-Verlag, Berlin, 1990), pp. 141–188.

E. C. Monahan, “The ocean as a source for atmospheric particles,” in The Role of Air-Sea Exchange in Geochemical Cycling, P. Buat-Menard, ed. (Reidel, Dordrecht, The Netherlands, 1986), pp. 129–163.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic plot of two representative bubble size distributions frequently found in the ocean. The figure is plotted such that the two distributions have the same mean radius of 50 μm. For the DI (dashed curve), the plateau is located between 30 and 50 μm, and for the DII (solid curve) the minimum radius is approximately 34 μm.

Fig. 2
Fig. 2

Variations of (a) scattering and (b) backscattering efficiency with bubble size and as a function of thickness of a protein film. To identify the variations for both small and large sizes, the same data set is plotted in logarithmic scale (left) and linear scale (right). The efficiencies for bubbles with a film of various thicknesses are indicated by the thickness (in micrometers), whereas clean denotes the efficiency for clean bubbles and protein refers to a homogeneous protein sphere of the same size as bubbles.

Fig. 3
Fig. 3

(a) Mean scattering and (b) backscattering efficiencies of clean bubbles as a function of mean bubble radius for DI and DII bubble size distributions. DI (solid curves) from left to right correspond to r a = 0, 0.2, 0.4, 0.6, 0.8, 1.0r b . Also shown as dotted curves are the efficiency factors of single clean bubbles for reference.

Fig. 4
Fig. 4

(a) Mean scattering and (b) backscattering efficiencies of coated bubbles as a function of mean bubble radius and film thickness. From top to bottom, the scattering efficiencies for various coatings are indicated by the legend in the same order. For backscattering, the curves corresponding to various film thickness or composition are indicated individually.

Fig. 5
Fig. 5

Variations of (a) mean scattering and (b) backscattering efficiencies of coated bubbles with mean bubble radius as a function of thickness and imaginary index of the films. The solid curves are for clean bubbles, dashed curves for coated bubbles with 0.1-μm-thick film and dotted curves for 1-μm-thick film. Within a group, the curves from top to bottom represent efficiencies for bubbles with films without absorption, of imaginary index of 0.001, and of imaginary index of 0.006, respectively.

Fig. 6
Fig. 6

(a) Scattering and (b) backscattering coefficients (solid lines) for clean bubble populations, represented by the generalized size distribution, as a function of mean bubble radius (in micrometers) (λ = 550 nm) for various bubble number densities. Also shown in numbers are the corresponding coefficients estimated with in situ observations (Table 1, column 1) based on the clean bubbles case. The two dashed lines are the upper and lower boundaries of coefficients calculated with Eq. (9) for Fig. 6(a) and Eq. (10) for Fig. 6(b) for [Chl] = 1 mg m-3. The two dotted lines are for [Chl] = 0.03 mg m-3.

Fig. 7
Fig. 7

Cumulative contributions (symbols) of bubbles and micro-organisms (dashed lines) compared with in situ observed (a) scattering and (b) backscattering coefficients, represented by two solid lines corresponding to upper and lower boundaries of Eqs. (9) and (10), respectively. The bubbles are assumed to be clean, and their number density varies from 106 m-3 (*) to 107 m-3 (○). The backscattering that is due to micro-organisms alone could not account for the in situ measurements of backscattering.

Fig. 8
Fig. 8

Cumulative contributions (stars) of bubbles (triangles) and micro-organisms (dashed lines) as compared with in situ observed (a) scattering and (b) backscattering coefficients, represented by two solid lines corresponding to upper and lower boundaries of Eqs. (9) and (10), respectively. The bubble number density varies with the chlorophyll concentration, N 0 = 106, 3 × 106, and 8 × 106 m-3 for [Chl] = 0.01–0.1, 0.1–1, and 1.0–10 mg m-3. The filled symbols correspond to the case in which the clean bubbles are replaced by dirty bubbles, which are assumed to increase the backscattering two fold according to Fig. 4.

Fig. 9
Fig. 9

(a) Ratio of enhanced diffuse reflectance R B that is due to clean bubble injection to the normal reflectance R as a function of [Chl] for various wavelengths. The upper three curves are estimates obtained with clean bubbles having a number density of 107 m-3, whereas the lower three use 106 m-3. (b) The variation of OC B after the bubble injection relative to OC before the bubble injection as a function of [Chl] for clean and coated bubbles (protein film of 0.1 μm). The upper two curves are for injected bubbles with a number density of 106 m-3, and the lower two use 107 m-3.

Tables (2)

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Table 1 Bubble Experiments and Their Statistical and Optical Characteristics

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Table 2 Mean Optical Efficiency Factors of Bubbles and Autotropic and Heterotrophic Plankton

Equations (10)

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R λ = f b b λ / a λ ,
j = r min r max   Q j r π r 2 n r d r ,
n r = N 0 p r m - 3   μ m - 1 ,
p r = c 1   r 4 0 r < r a c 2 r a r < r b c 3   r - 4 r b r ,
p r = c   r - 4   r r 0 ,
j = N 0 Q j ¯   s ¯ ,
Q j ¯ = r min r max   Q j r p r π r 2 d r r min r max   p r π r 2 d r ,
s ¯ = r min r max   p r π r 2 d r .
b λ = 550 λ 0.30 ± 0.15 Chl 0.62 - b w λ ,
b b λ = b λ 2 × 10 - 3 + 2 × 10 - 2 × 0.5 - 0.25   log 10 Chl 550 λ ,

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