Abstract

Irradiance fluctuations in the natural underwater light field close to the sea surface have an amplitude that is of the same order of magnitude as the mean irradiance. A principal source of these fluctuations, differential refraction by surface waves, is examined in this paper, and some experimental data obtained at an experimental site in the Bight of Abaco, Bahamas, are presented. A first-order single-ray theory is developed and the predictions of this theory are compared with the data. The theory identifies five refractive effects, the most important of which is the focusing and defocusing of light beams by fluctuations of surface curvature.

© 1970 Optical Society of America

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  1. The reader is forewarned of an unavoidable duplication of terminology in the optical and geophysical aspects of this paper. Some confusion may result if this duplication is not borne in mind. Here, the terms “frequency” and “wavenumber” refer to the geophysical distribution of irradiance in the sea, not to the wave-mechanical properties of light.
  2. Y. Le Grand, Ann. Inst. Oceanogr. 19, 393 (1939).
  3. W. V. Burt, J. Meteorol. 11, 283 (1959).
    [Crossref]
  4. C. Cox and W. Munk, Bull. Scripps Inst. Oceanogr. 6, 401 (1956).
  5. K. Hishida and M. Kishino, J. Oceanogr. Soc. Japan 27, 748 (1965).
  6. N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).
  7. Y. Mullamaa, Izv. Akad. Nauk. USSR. Sev. Geofiz. 8, 1232 (1964).
  8. H. Schenck, J. Opt. Soc. Am. 47, 653 (1957).
    [Crossref]
  9. J. Dera and H. R. Gordon, Linmnol. Oceanogr. 13, 697 (1968).
    [Crossref]
  10. J. Dera and J. Olszewski, Acta Geophys. Polon. 15, 351 (1967).
  11. In this paper, the terms “spectrum” and “spectra” are not optical terms, but refer to a time-series analysis of a random fluctuation.
  12. z is taken positive upwards with z= 0 at the mean water surface.
  13. Boldface letters will denote two-dimensional (horizontal) vector quantities. (h,v) will denote a three-dimensional vector quantity with horizontal projection h and vertical component v.
  14. This calculation assumes that the mean irradiance field is entirely the result of direct sun rays. The observations of Dera and Olszewski suggest that this assumption is only crudely satisfied. The value of the resulting attenuation coefficient is probably more representative of a downwelling-irradiance attenuation coefficient than it is of a beam-attenuation coefficient.
  15. As evidenced in the coordinate transformation (3) in Sec. III.
  16. The expansion parameter is essentially the rms surface-wave slope.
  17. The variables ξ and η here have a meaning different from their previous meaning.

1968 (1)

J. Dera and H. R. Gordon, Linmnol. Oceanogr. 13, 697 (1968).
[Crossref]

1967 (1)

J. Dera and J. Olszewski, Acta Geophys. Polon. 15, 351 (1967).

1965 (1)

K. Hishida and M. Kishino, J. Oceanogr. Soc. Japan 27, 748 (1965).

1964 (1)

Y. Mullamaa, Izv. Akad. Nauk. USSR. Sev. Geofiz. 8, 1232 (1964).

1959 (1)

W. V. Burt, J. Meteorol. 11, 283 (1959).
[Crossref]

1957 (1)

1956 (1)

C. Cox and W. Munk, Bull. Scripps Inst. Oceanogr. 6, 401 (1956).

1939 (1)

Y. Le Grand, Ann. Inst. Oceanogr. 19, 393 (1939).

Burt, W. V.

W. V. Burt, J. Meteorol. 11, 283 (1959).
[Crossref]

Cox, C.

C. Cox and W. Munk, Bull. Scripps Inst. Oceanogr. 6, 401 (1956).

Dera, J.

J. Dera and H. R. Gordon, Linmnol. Oceanogr. 13, 697 (1968).
[Crossref]

J. Dera and J. Olszewski, Acta Geophys. Polon. 15, 351 (1967).

Gordon, H. R.

J. Dera and H. R. Gordon, Linmnol. Oceanogr. 13, 697 (1968).
[Crossref]

Hishida, K.

K. Hishida and M. Kishino, J. Oceanogr. Soc. Japan 27, 748 (1965).

Jerlov, N. G.

N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).

Kishino, M.

K. Hishida and M. Kishino, J. Oceanogr. Soc. Japan 27, 748 (1965).

Le Grand, Y.

Y. Le Grand, Ann. Inst. Oceanogr. 19, 393 (1939).

Mullamaa, Y.

Y. Mullamaa, Izv. Akad. Nauk. USSR. Sev. Geofiz. 8, 1232 (1964).

Munk, W.

C. Cox and W. Munk, Bull. Scripps Inst. Oceanogr. 6, 401 (1956).

Olszewski, J.

J. Dera and J. Olszewski, Acta Geophys. Polon. 15, 351 (1967).

Schenck, H.

Acta Geophys. Polon. (1)

J. Dera and J. Olszewski, Acta Geophys. Polon. 15, 351 (1967).

Ann. Inst. Oceanogr. (1)

Y. Le Grand, Ann. Inst. Oceanogr. 19, 393 (1939).

Bull. Scripps Inst. Oceanogr. (1)

C. Cox and W. Munk, Bull. Scripps Inst. Oceanogr. 6, 401 (1956).

Izv. Akad. Nauk. USSR. Sev. Geofiz. (1)

Y. Mullamaa, Izv. Akad. Nauk. USSR. Sev. Geofiz. 8, 1232 (1964).

J. Meteorol. (1)

W. V. Burt, J. Meteorol. 11, 283 (1959).
[Crossref]

J. Oceanogr. Soc. Japan (1)

K. Hishida and M. Kishino, J. Oceanogr. Soc. Japan 27, 748 (1965).

J. Opt. Soc. Am. (1)

Linmnol. Oceanogr. (1)

J. Dera and H. R. Gordon, Linmnol. Oceanogr. 13, 697 (1968).
[Crossref]

Other (9)

The reader is forewarned of an unavoidable duplication of terminology in the optical and geophysical aspects of this paper. Some confusion may result if this duplication is not borne in mind. Here, the terms “frequency” and “wavenumber” refer to the geophysical distribution of irradiance in the sea, not to the wave-mechanical properties of light.

N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).

In this paper, the terms “spectrum” and “spectra” are not optical terms, but refer to a time-series analysis of a random fluctuation.

z is taken positive upwards with z= 0 at the mean water surface.

Boldface letters will denote two-dimensional (horizontal) vector quantities. (h,v) will denote a three-dimensional vector quantity with horizontal projection h and vertical component v.

This calculation assumes that the mean irradiance field is entirely the result of direct sun rays. The observations of Dera and Olszewski suggest that this assumption is only crudely satisfied. The value of the resulting attenuation coefficient is probably more representative of a downwelling-irradiance attenuation coefficient than it is of a beam-attenuation coefficient.

As evidenced in the coordinate transformation (3) in Sec. III.

The expansion parameter is essentially the rms surface-wave slope.

The variables ξ and η here have a meaning different from their previous meaning.

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Figures (7)

Fig. 1
Fig. 1

Mean irradiance as a function of depth. Presentation of the data is suggested by zero-order theory. Averages were calculated over a 4-min section of each run. + values were observed while the irradiance meter was being lowered and o values were observed while it was being raised.

Fig. 2
Fig. 2

Irradiance fluctuations during the first 80 s of run 46.

Fig. 3
Fig. 3

Energy spectra of irradiance fluctuations (upper panel) and surface elevation (lower panel) for runs 46–52. Spectra are functions of radial frequency ω. Wave spectra are plotted on two scales. The Nyquist frequency Ω is 100.53 s−1 for the irradiance spectra and for the larger-scale wave spectra and is 6.2832 s−1 for the smaller-scale wave spectra. 90% confidence limits are 0.96 to 1.18 for the irradiance spectra and 0.59 to 2.1 for the wave spectra. Spectra were calculated with a Lanczos squared data window using FESTSA (Holsten and Groves).

Fig. 4
Fig. 4

Ratio between the irradiance fluctuation variance and the square of the mean irradiance as a function of depth. The curve shown is a best fit to the experimental points.

Fig. 5
Fig. 5

Geometry of the ray theory. R and Q lie in horizontal planes. S lies in the water surface. R is the ray projection of S, and Q is the vertical projection of S. (l,lz) is a unit vector in the direction of the sun, (m,mz) is a unit vector pointing along the refracted ray with the same sense as (l,lz) and (n,nz) is the unit normal to the water surface.

Fig. 6
Fig. 6

Ratio between the irradiance fluctuation spectra and the wave spectra for runs 46–49. The Nyquist frequency Ω is 6.2832 s−1. Confidence limits are 0.59 to 2.1 for all spectra, giving a proportional variance of 0.2 or a proportional deviation of 0.45 for the ratio. The + values are from experiment, and the curves are calculated from first-order theory.

Fig. 7
Fig. 7

Ratio between the irradiance fluctuation spectra and the wave spectra for runs 49–52. Explanation of symbols identical with Fig. 6.

Tables (2)

Tables Icon

Table I Experimental parameters. Depth figures have been corrected for tide. Absolute depth is accurate to about 10 cm, relative depths to about 3 cm. Azimuths are measured clockwise from north.

Tables Icon

Table II Orders of magnitude of various terms in Eq. (5) for a progressive wave.

Equations (27)

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F F 0 ( z ) = I 0 T 0 l z exp [ α B ( l ) z ] ,
I ( l , x , z , t ) = - I 0 l ,
I z ( l , x , z , t ) = - I 0 l z ,
λ ( n , n z ) × ( l , l z ) = ( n , n z ) × ( m , m z ) ,
( n , n z ) { - ζ [ 1 + ( ζ ) 2 ] 1 2 , 1 [ 1 + ( ζ ) 2 ] 1 2 }
y = x - ( m / m z ) ( ζ - z ) .
F ( l , η , z , t ) = 1 A R ( η ) d 2 y I ( l , y , z , t ) m z ( l , y , z , t ) ,
F = 1 A R d 2 y I m z = I 0 A S d S μ T ( μ ) exp [ - α ( ζ - z ) m z ] = I 0 A Q d 2 x μ n z T ( μ ) exp [ - α ( ζ - z ) m z ] .
F = I 0 A d 2 y J μ n z T ( μ ) exp [ - α ( ζ - z ) m z ] ,
J ( l , y , z , t ) | x 1 y 1 x 2 y 1 x 1 y 2 x 2 y 2 |
x = x ( l , y , z , t ) .
ζ = ζ 0 + ζ 1 + n = n 0 + n 1 + n z = n z 0 + n z 1 +                         F = F 0 + F 1 + .
n 0 = 0 , n z 0 = 1 , m 0 = μ l , m z 0 = B ( l ) - 1 , x 0 = y - μ B ( l ) l z , J 0 = 1 , T 0 = T ( l z ) ,
F 0 = I 0 T 0 l z exp [ α B ( l ) z ] ,
B ( l ) ( 1 - λ 2 l 2 ) - 1 2 .
n 1 = - ζ 1 , n z 1 = 0 , m 1 = - [ B ( l ) - 1 - λ l z ] ζ 1 , m z 1 = λ G ( l ) l · ζ 1 , x 1 = λ B ( l ) l ζ 1 + G ( l ) z ζ 1 + H ( l ) l z l · ζ 1 , J 1 = λ B ( l ) l · ζ 1 + G ( l ) z 2 ζ 1 + H ( l ) z l · ( l · ζ 1 ) , T 1 = - ( d T / d μ ) ( l z ) l · ζ 1 ,
F 1 = F 0 A R d 2 y [ - α B ( l ) ζ 1 - α C ( l ) z l · ζ 1 - D ( l ) l · ζ 1 + G ( l ) z 2 ζ 1 + H ( l ) z l · ( l · ζ 1 ) ] ,
C ( l ) λ [ B ( l ) - 1 - λ l z ] B ( l ) 3 = λ B ( l ) 2 G ( l ) , D ( l ) T ( l z ) - 1 ( d T / d μ ) ( l z ) + l z - 1 - λ B ( l ) - 1 , G ( l ) [ B ( l ) - 1 - λ l z ] B ( l ) ,
H ( l ) λ 2 [ B ( l ) - 1 - λ l z ] B ( l ) 3 = λ C ( l ) ,
F 1 ( η , t ) = F 0 { - α B ( l ) ζ 1 ( ξ , t ) - α C ( l ) z l · ζ 1 ( ξ , t ) - D ( l ) l · ζ 1 ( ξ , t ) + G ( l ) z 2 ζ 1 ( ξ , t ) + H ( l ) z l · [ l · ζ 1 ( ξ , t ) ] } ,
ξ ( l , η , z ) η - B ( l ) l z .
C F 1 2 ( η , τ ) F 1 ( y , t ) F 1 ( y + η , t + τ )
C ζ 1 2 ( ξ , τ ) ζ 1 ( x , t ) ζ 1 ( x + ξ , t + τ )
C F 1 2 ( η , t ) = d 2 k E F 1 2 ( k ) cos [ k · η - ω ( k , h ) τ ]
C ζ 1 2 ( ξ , τ ) = d 2 k E ζ 1 2 ( k ) cos [ k · ξ - ω ( k , h ) τ ] ,
E F 1 2 ( k ) = F 0 2 { [ α B ( l ) + G ( l ) z k 2 + H ( l ) z ( l · k ) 2 ] 2 + ( l · k ) 2 [ D ( l ) + α C ( l ) z ] 2 } E ζ 1 2 ( k ) .
F 1 2 = - d 2 k E F 1 2 ( k ) = ( I + J z + K z 2 ) F 0 ( z ) 2 ,