And time of year

At any given point on the Earth's surface the daylength and the solar elevation reach their maximum values in the summer and their minimum values in the winter. Substituting into eqn 2.7 we can obtain expressions for the noon solar elevation on the longest summer day (d = 23 °27')

as a function of latitude. At the latitude of Canberra (35 ° S) for example, the corresponding solar elevations are 78.5 and 31.5 °: at the latitude of London (51.5 ° N) they are 62 and 15 °.

Maximum and minimum daylengths can also be obtained using eqn 2.7. If the time (expressed as an angle) at sunrise is ts then (since sin b = 0 at sunrise) at any time of year cos ts = tan g tan d (2.14)

Daylength expressed as an angle is (360 ° - 2 ts), which is equal to 2 cos—1 (— tan g tan d). Expressed in hours, daylength is given by

The longest day is therefore 0.133cos—1 (—0.43377tang) h and the shortest day is 0.133cos—1 (0.43377tang)h.

With increasing latitude, the solar elevation at noon and therefore the maximum value of solar irradiance (Em) decrease, which in accordance with eqn 2.11, decreases the daily insolation. In the summer, however, this effect is counteracted by the increase in daylength (N in eqn 2.11) with increasing latitude, and the net result is that high-latitude regions can have a slightly greater daily insolation in midsummer than tropical regions. In the winter, of course, the high-latitude regions have shorter days (zero daylengths in the polar regions) as well as lower solar elevations, so their daily insolation is much less than at low latitudes. Throughout most of the year, in fact, the rule-of-thumb holds that the higher the latitude, the lower the daily insolation. Figure 2.8 shows the change in daily insolation throughout the year, calculated ignoring atmospheric losses, for a range of latitudes. Figure 2.9 shows the true (measured) daily insolation as a function of time of year at a southern hemisphere site, near Canberra, Australia, averaged over a three-year period, illustrating the steady rise to a peak in the summer followed by a decline to the winter minimum. These data, unlike those in Fig. 2.8, include the effects of cloud and atmospheric haze.

In oceanic regions with monsoonal climates there can be two seasonal peaks of intensity of solar radiation within the year.168a,3° Figure 2.10 shows the mean monthly surface-incident PAR over a seven-year period averaged over the

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

Fig. 2.8 Change in calculated daily insolation (ignoring the influence of the atmosphere) throughout the year at different latitudes in the northern hemisphere. The latitude is indicated above each curve. Plotted from data of Kondratyev (1954).

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

Fig. 2.8 Change in calculated daily insolation (ignoring the influence of the atmosphere) throughout the year at different latitudes in the northern hemisphere. The latitude is indicated above each curve. Plotted from data of Kondratyev (1954).

Arabian Sea. After the winter minimum in December to January, Ed(PAR) rises to a spring maximum in March, April and May, and then declines to a minimum in June and July in response to increased cloud cover and aerosols associated with the onset of the southwest monsoon.30 Solar irradiance then rises to a second, lower, peak in September, following which it declines once again to the winter minimum. In more typical parts of the ocean, such as the North Atlantic, there is a single irradiance maximum, in the summer.

2.5 Transmission across the air-water interface

To become available in the aquatic ecosystem, the solar radiation that has penetrated through the atmosphere must now find its way across the air-water interface. Some of it will be reflected back into the atmosphere.

Fig. 2.9 Change in measured daily insolation throughout the year at the CSIRO Ginninderra Experiment Station, (35° S, 149° E). The curve corresponds to the daily average for each calendar month, averaged over the period 1978-80.

Monthly mean 1979-85

Fig. 2.10 The mean interannual irradiance of PAR (Wm~2) at the sea surface for the entire Arabian Sea basin (excluding the Persian Gulf and the Red Sea) from 1979 to 1985. Plotted from data in Plate 4 of Arnone et al. (1998). J. Geophys. Res., 103, 7735-48.

The proportion of the incident light that is reflected by a flat water surface increases from 2% for vertically incident light towards 100% as the beam approaches grazing incidence. The dependence of reflectance, r, of

100 r

10 16

10 16

Angle of incidence/observation

Fig. 2.11 Reflectance of water surface as a function of zenith angle of light (incident from above), at different wind speeds (data of Gordon, 1969; Austin, 1974a).

unpolarized light on the zenith angle of the incident light in air (0a), and on the angle to the downward vertical of the transmitted beam in water, (0w), is given by Fresnel's Equation

1 sin2 {Pa - 0w) , 1 tan2 (0a - Qw) n r = -z —o--hx -T7-r (2.16

The angle, 0w, in water is itself determined by 0a, and the refractive index, as we shall shortly see. The percentage reflectance from a flat water surface as a function of zenith angle is shown in Fig. 2.11, and in tabular form in Table 2.1. It will be noted that reflectance remains low, increasing only slowly, up to zenith angles of about 50 but rises very rapidly thereafter.

Roughening of the water surface by wind has little effect on the reflectance of sunlight from high solar elevations. At low solar elevations, on the other hand, reflectance is significantly lowered by wind since the roughening of the surface on average increases the angle between the light direction and the surface at the point of entry. The three lower curves in Fig. 2.11 show the effect of wind at different velocities on the reflectance.494,41

Table 2.1 Reflectance of unpolarized light from aflat water surface. The values of reflectance have been calculated using eqns 2.16 and 2.19, assuming that the water has a refractive index of 1.33.

Zenith angle of

Reflectance

Zenith angle of

Reflectance

incidence 6a (degrees)

(%)

incidence 6a (degrees)

(%)

0.0

2.0

50.0

3.3

5.0

2.0

55.0

4.3

10.0

2.0

60.0

5.9

15.0

2.0

65.0

8.6

20.0

2.0

70.0

13.3

25.0

2.1

75.0

21.1

30.0

2.1

80.0

34.7

35.0

2.2

85.0

58.3

40.0

2.4

87.5

76.1

45.0

2.8

89.0

89.6

As wind speed increases the waves begin to break, and whitecaps are formed. The fraction of the ocean surface covered by whitecaps, W, is a function of windspeed, U (ms—1 at 10 m above surface), and can be represented as a power law

The coefficient A and the exponent B are functions of water temperature, but combining data from a range of water temperatures, Spillane and Doyle (1983) found the relationship

to give the best fit. Equation 2.18 predicts, for example, that whitecaps cover about 1% of the sea surface at a wind speed of 10 m s—1, and 13% at 25ms—1. Freshly formed whitecaps consist of many layers of bubbles, and have a reflectance of about 55%.1464,1284 They have, however, a lifetime of only 10 to 20 s in the ocean, and Koepke (1984) found that as they decay their reflectance decreases markedly due to thinning of the foam, and he estimated the effective reflectance to be on average only about 22%, and their contribution to total oceanic reflectance to be minor. Using eqn 2.18 we can calculate the additional ocean surface reflectance due to whitecaps to be only about 0.25% at a wind speed of 10m s—1, and ~3% at 25m s—1.

Because of the complex angular distribution of skylight, the extent to which it is reflected by a water surface is difficult to determine. If the very

Fig. 2.12 Refraction and reflection of light at air-water boundary. (a) A light beam incident from above is refracted downwards within the water: a small part of the beam is reflected upwards at the surface. (b) A light beam incident from below at a nadir angle of 40 ° is refracted away from the vertical as it passes through into the air: a small part of the beam is reflected downwards again at the water-air boundary. (c) A light beam incident from below at a nadir angle greater than 49 ° undergoes complete internal reflection at the water-air boundary.

Fig. 2.12 Refraction and reflection of light at air-water boundary. (a) A light beam incident from above is refracted downwards within the water: a small part of the beam is reflected upwards at the surface. (b) A light beam incident from below at a nadir angle of 40 ° is refracted away from the vertical as it passes through into the air: a small part of the beam is reflected downwards again at the water-air boundary. (c) A light beam incident from below at a nadir angle greater than 49 ° undergoes complete internal reflection at the water-air boundary.

approximate assumption is made that the radiance is the same from all directions, then a reflectance of 6.6% for a flat water surface is obtained.636 For an incident radiance distribution such as might be obtained under an overcast sky, the reflectance is calculated to be about 5.2%.1074 Roughening of the surface by wind will lower reflectance for the diffuse light from a clear or an overcast sky.

As the unreflected part of a light beam passes across the air-water interface, it changes its direction to the vertical (while remaining, if the surface is flat, in the same vertical plane), due to refraction. The phenomenon of refraction is the result of the differing velocities of light in the two media, air and water. The change in angle (Fig. 2.12) is governed by Snell's Law sin^ = nw (2.19)

sin 0W na where nw and na are the refractive indices of water and air, respectively. The ratio of the refractive index of water to that of air is a function of temperature, salt concentration and the wavelength of the light in question. For our purposes a value of 1.33 for nw/na is close enough for both sea and fresh water at normal ambient temperatures and for light of any wavelength within the photosynthetic range. As Fig. 2.12 shows, the effect of refraction is to move the light closer to the vertical in the aquatic medium than it was in air. It will be noted that even light reaching the surface at grazing incidence (0a approaching 90 °) is refracted downwards so that 0W should not be greater than about 49 ° for a calm surface.

When the water is disturbed, some of the light will be found at angles greater than 49 ° after passing through the surface: nevertheless most of the downwelling light will be at values of 0w between 0 ° and 49 Statistical relations between the distribution of surface slopes and wind speed were derived by Cox and Munk (1954) from aerial photographs of the Sun's glitter pattern. The Cox and Munk data are summarized by the equations gu2 = 0.00316 U ± 0.004 (2.20)

where U is the wind speed at 12.5m above the water surface, su2 is the mean square slope of the waves measured in the upwind/downwind (i.e. parallel to the wind) direction, and sc2 is the crosswind value, measured at right angles to the wind. Wave slope in this context is tan C, where C is the angle between the vertical and the normal to the sea surface at a given point. Calculations using these relations show that, as might be expected, as wind speed increases, the underwater light becomes more diffuse.515 Recent measurements of wind speed and the associated ocean sunglint intensity and pattern from space, using instruments on the ADEOS satellite, are in excellent agreement with the Cox and Munk slope distribution model.158

Snell's Law works in reverse also. A light beam passing upward within the water at angle 0w to the downward vertical will emerge through the calm surface into the air at angle 0a to the zenith, in accordance with eqn 2.19. There is, however, a very important difference between transmission from water to air, and from air to water, namely that in the former case complete internal reflection can occur. If, in the case of the upward-directed light beam within the water, 0w is greater than 49 ° then all the light is reflected down again by the water-air interface. Disturbance of the surface by wind decreases water-to-air transmission for upwelling light at angles within the range 0w = 0 ° to 0w = 49 ° but, by ensuring that not all the light is internally reflected, it increases transmission within the range 0w = 49 ° to 0w = 90

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