If we are to understand the ways in which the prevailing light field changes with depth in a water body, then we must first consider what are the essential attributes of a light field in which changes might be anticipated. The definitions of these attributes, in part, follow the report of the Working Groups set up by the International Association for the Physical Sciences of the Ocean (1979), but are also influenced by the more fundamental analyses given by Preisendorfer (1976). A more recent account of the definitions and concepts used in hydrologic optics is that by Mobley (1994).

We shall generally express direction within the light field in terms of the zenith angle, 0 (the angle between a given light pencil, i.e. a thin parallel beam, and the upward vertical), and the azimuth angle, f (the angle between the vertical plane incorporating the light pencil and some other specified vertical plane such as the vertical plane of the Sun). In the case of the upwelling light stream it will sometimes be convenient to express a direction in terms of the nadir angle, 0n (the angle between a given light pencil and the downward vertical). These angular relations are illustrated in Fig. 1.2.

Radiant flux, F, is the time rate of flow of radiant energy. It may be expressed in W (Js-1) or quanta s-1.

Radiant intensity, I, is a measure of the radiant flux per unit solid angle in a specified direction. The radiant intensity of a source in a given direction is the radiant flux emitted by a point source, or by an element of an extended source, in an infinitesimal cone containing the given direction, divided by that element of solid angle. We can also speak of radiant intensity at a point in space. This, the field radiant intensity, is the radiant flux at that point in a specified direction in an infinitesimal cone

Fig. 1.2 The angles defining direction within a light field. The figure shows a downward and an upward pencil of light, both, for simplicity, in the same vertical plane. The downward pencil has zenith angle 0; the upward pencil has nadir angle 0n, which is equivalent to a zenith angle of (180° - 0n). Assuming the xy plane is the vertical plane of the Sun, or other reference vertical plane, then 0 is the azimuth angle for both light pencils.

Fig. 1.2 The angles defining direction within a light field. The figure shows a downward and an upward pencil of light, both, for simplicity, in the same vertical plane. The downward pencil has zenith angle 0; the upward pencil has nadir angle 0n, which is equivalent to a zenith angle of (180° - 0n). Assuming the xy plane is the vertical plane of the Sun, or other reference vertical plane, then 0 is the azimuth angle for both light pencils.

containing the given direction, divided by that element of solid angle. I has the units W (or quanta s_1)steradian_1.

If we consider the radiant flux not only per unit solid angle but also per unit area of a plane at right angles to the direction of flow, then we arrive at the even more useful concept of radiance, L. Radiance at a point in space is the radiant flux at that point in a given direction per unit solid angle per unit area at right angles to the direction of propagation. The meaning of this field radiance is illustrated in Figs. 1.3a and b. There is also surface radiance, which is the radiant flux emitted in a given direction per unit solid angle per unit projected area (apparent unit area, seen from the viewing direction) of a surface: this is illustrated in Fig. 1.3c. To indicate that it is a function of direction, i.e. of both zenith and azimuth angle, radiance is commonly written as L(0, f). The angular structure of a light field is expressed in terms of the variation of radiance with 0 and f. Radiance has the units W (or quanta s_1)m_2steradian_1.

dS cos0 dffl

Fig. 1.3 Definition of radiance. (a) Field radiance at a point in space. The field radiance at P in the direction D is the radiant flux in the small solid angle surrounding D, passing through the infinitesimal element of area dA at right angles to D divided by the element of solid angle and the element of area. (b) Field radiance at a point in a surface. It is often necessary to consider radiance at a point on a surface, from a specified direction relative to that surface. dS is the area of a small element of surface. L(0, 0) is the radiance incident on dS at zenith angle d (relative to the normal to the surface) and azimuth angle 0: its value is determined by the radiant flux directed at dS within the small solid angle, do, centred on the line defined by d and 0. The flux passes perpendicularly across the area dS cos d, which is the projected area of the element of surface, dS, seen from the direction d, 0. Thus the radiance on a point in a surface, from a given direction, is the radiant flux in the specified direction per unit solid angle per unit projected area of the surface. (c) Surface radiance. In the case of a surface that emits radiation the intensity of the flux leaving the surface in a specified direction is expressed in terms of the surface radiance, which is defined in the same way as the field radiance at a point in a surface except that the radiation is considered to flow away from, rather than on to, the surface.

Irradiance (at a point of a surface), E, is the radiant flux incident on an infinitesimal element of a surface, containing the point under consideration, divided by the area of that element. Less rigorously, it may be defined as the radiant flux per unit area of a surface.* It has the units Wm~2 or quanta (or photons) s_1m~2, or mol quanta (or photons) s_1m~2, where 1.0 mol photons is 6.02 x 1023 (Avogadro's number) photons. One mole of photons is sometimes referred to as an einstein, but this term is now rarely used.

* Terms such as 'fluence rate' or 'photon fluence rate', sometimes to be found in the plant physiological literature, are superfluous and should not be used.

Downward irradiance, Ed, and upward irradiance, Eu, are the values of the irradiance on the upper and the lower faces, respectively, of a horizontal plane. Thus, Ed is the irradiance due to the downwelling light stream and Eu is that due to the upwelling light stream.

The relation between irradiance and radiance can be understood with the help of Fig. 1.3b. The radiance in the direction defined by 0 and f is L (0, f) W (or quanta s—1) per unit projected area per steradian (sr). The projected area of the element of surface is dS cos 0 and the corresponding element of solid angle is do. Therefore the radiant flux on the element of surface within the solid angle dm is L(0, f)dS cos 0 do. The area of the element of surface is dS and so the irradiance at that point in the surface where the element is located, due to radiant flux within do, is L(0, f) cos 0 do. The total downward irradiance at that point in the surface is obtained by integrating with respect to solid angle over the whole upper hemisphere

The total upward irradiance is related to radiance in a similar manner except that allowance must be made for the fact that cos 0 is negative for values of 0 between 90 and 180 °

Alternatively the cosine of the nadir angle, 0n (see Fig. 1.2), rather than of the zenith angle, may be used

The —2p subscript is simply to indicate that the integration is carried out over the 2p sr solid angle in the lower hemisphere.

The net downward irradiance, E, is the difference between the downward and the upward irradiance

It is related to radiance by the eqn

which integrates the product of radiance and cos 0 over all directions: the fact that cos 0 is negative between 90 and 180 ° ensures that the contribution of upward irradiance is negative in accordance with eqn 1.8. The net downward irradiance is a measure of the net rate of transfer of energy downwards at that point in the medium, and as we shall see later is a concept that can be used to arrive at some valuable conclusions.

The scalar irradiance, E0, is the integral of the radiance distribution at a point over all directions about the point

Scalar irradiance is thus a measure of the radiant intensity at a point, which treats radiation from all directions equally. In the case of irradi-ance, on the other hand, the contribution of the radiation flux at different angles varies in proportion to the cosine of the zenith angle of incidence of the radiation: a phenomenon based on purely geometrical relations (Fig. 1.3, eqn 1.5), and sometimes referred to as the Cosine Law. It is useful to divide the scalar irradiance into a downward and an upward component. The downward scalar irradiance, E0d, is the integral of the radiance distribution over the upper hemisphere

The upward scalar irradiance is defined in a similar manner for the lower hemisphere

Scalar irradiance (total, upward, downward) has the same units as irradiance.

It is always the case in real-life radiation fields that irradiance and scalar irradiance vary markedly with wavelength across the photosyn-thetic range. This variation has a considerable bearing on the extent to which the radiation field can be used for photosynthesis. It is expressed in terms of the variation in irradiance or scalar irradiance per unit spectral distance (in units of wavelength or frequency, as appropriate) across the spectrum. Typical units would be W (or quantas-1)m-2nm-1.

If we know the radiance distribution over all angles at a particular point in a medium then we have a complete description of the angular structure of the light field. A complete radiance distribution, however, covering all zenith and azimuth angles at reasonably narrow intervals, represents a large amount of data: with 5 ° angular intervals, for example, the distribution will consist of 1369 separate radiance values. A simpler, but still very useful, way of specifying the angular structure of a light field is in the form of the three average cosines - for downwelling, upwelling and total light - and the irradiance reflectance.

The average cosine for downwelling light, , at a particular point in the radiation field, may be regarded as the average value, in an infini-tesimally small volume element at that point in the field, of the cosine of the zenith angle of all the downwelling photons in the volume element. It can be calculated by summing (i.e. integrating) for all elements of solid angle (do) comprising the upper hemisphere, the product of the radiance in that element of solid angle and the value of cos 0 (i.e. L(0, f) cos 0), and then dividing by the total radiance originating in that hemisphere. By inspection of eqns 1.5 and 1.11 it can be seen that i.e. the average cosine for downwelling light is equal to the downward irradiance divided by the downward scalar irradiance. The average cosine for upwelling light, ~pu, may be regarded as the average value of the cosine of the nadir angle of all the upwelling photons at a particular point in the field. By a similar chain of reasoning to the above, we conclude that ~pu is equal to the upward irradiance divided by the upward scalar irradiance

In the case of the downwelling light stream it is often useful to deal in terms of the reciprocal of the average downward cosine, referred to by Preisendorfer (1961) as the distribution function for downwelling light, Dd, which can be shown712 to be equal to the mean pathlength per vertical metre traversed, of the downward flux of photons per unit horizontal area per second. Thus Dd = 1 /^d. There is, of course, an analogous distribution function for the upwelling light stream, defined by Du = 1/~pu.

The average cosine, for the total light at a particular point in the field may be regarded as the average value, in an infinitesimally small volume element at that point in the field, of the cosine of the zenith angle of all the photons in the volume element. It may be evaluated by integrating the product of radiance and cos 0 over all directions and dividing by the total

radiance from all directions. By inspection of eqns 1.8, 1.9 and 1.10, it can be seen that the average cosine for the total light is equal to the net downward irradiance divided by the scalar irradiance

That Ed - Eu should be involved (rather than, say, Ed + Eu) follows from the fact that the cosine of the zenith angle is negative for all the upwelling photons (90 ° < 0 < 180 °). Thus a radiation field consisting of equal numbers of downwelling photons at 0 = 45 ° and upwelling photons at 0 = 135 ° would have p = 0.

Average cosine is often written as p(z) to indicate that it is a function of the local radiation field at depth z. The total radiation field present in the water column also has an average cosine, pc, this being the average value of the cosine of the zenith angle of all the photons present in the water column at a given time.716 In principle it could be evaluated by multiplying the value of p(z) in each depth interval by the proportion of the total water column radiant energy occurring in that depth interval, and then summing to obtain the average cosine for the whole water column, i.e. we would be making use of the relationship

where U(z) is the radiant energy density at depth z. The radiant energy density at depth z is equal to the scalar irradiance at that depth divided by the speed of light in water, cw

Making use of the fact that p(z) at any depth is equal to the net downward irradiance divided by the scalar irradiance (eqn 1.15), then substituting for p(z) and U(z) in eqn 1.16 and cancelling out, we obtain

Eo(z)dz

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