The preceding sections were fairly general, applicable to all theories of radiative transfer. To proceed further we have to make some specific assumptions, and the simplest one is that the radiation field consists of irradiances F in two and only two directions (streams), denoted as upward and downward. This is an idealization given that strictly monodirectional irradiances do not exist; even a laser beam has a finite angular spread. Scattering can therefore occur in only these two directions: a photon directed downward can be scattered only downward or upward, and similarly for a photon directed upward. We also ignore the polarization state of the
Figure 5.10: Downward (J.) and upward (|) irradiances F are different at z and z + Az because of absorption and scattering within Az. Note that the positive z-axis is downward.
radiation, which here makes no difference because the polarization state of light scattered by a spherically symmetric medium is not changed upon scattering in the forward and backward directions (see Sec. 7.4).
Conservation of downward radiant energy in a thin (relative to the mean free path) layer between z and z + Az (Fig. 5.10) yields
Fl(z + Az) = Fl(z) - KAzFl(z) - f3AzpuFl(z) + f3AzpnF^(z + Az). (5.36)
This is the mathematical form of the following statement. At the bottom of the layer the downward radiation is that incident at the top decreased by absorption (kAz is the probability of absorption) and by scattering upward in Az, but increased because upward radiation incident at the bottom of the layer is scattered downward in Az. We can ignore the probability that a photon is scattered more than once in Az (which depends on higher powers of f3Az) if we take 3Az C 1. The quantity p^ is the (conditional) probability that given that a downward photon is scattered, it is scattered in the upward direction, and similarly for pff. Divide both sides of Eq. (5.36) by Az and take the limit as Az ^ 0:
A similar radiant energy conservation argument applied to the upward irradiance yields dF
The sign reversal is a consequence of attenuation of upward radiation in the direction of decreasing z.
It is the presence of the third term on the right sides of Eqs. (5.37) and (5.38) that makes radiative transfer non-trivial. Were it not for this term, the solutions to both equations would be simple exponentials. What complicates matters is that downward radiation is a source for upward radiation and vice versa: the upward and downward irradiances are coupled.
Because photons can be scattered only upward or downward the sum of probabilities must satisfy p.iî + pu = pn + pîî = 1
By using Eq. (5.39) we can rewrite Eqs. (5.37) and (5.38) as dF\
These equations are easier to interpret. The first term on the right side expresses all the ways radiation can be removed from a particular direction (expenses), the second term all the ways radiation can be added to that direction (income). If you can balance a checkbook you can understand these equations.
Now we further assume that the medium is isotropic: p^ = p^, pn = p^. Such a medium is rotationally symmetric: rotate it by any amount and you can't tell that it has been rotated. A suspension of spherically symmetric scatterers is an isotropic medium, as is a suspension of randomly oriented asymmetric scatterers. An example of an anisotropic medium is a suspension of oriented, asymmetric scatterers.
We have to take some care with the term isotropic because it is used in different ways, sometimes in the same breath. An isotropic radiation field here is defined by F^ = F^; isotropic scattering by p ^ = p ^. Isotropic scattering does not necessarily imply an isotropic radiation field and conversely. To make matters more confusing, there are no isotropic scatter-ers of electromagnetic waves (acoustic waves are another story) in nature in the sense that they scatter equally in all directions (see Sec. 7.3). You can't beg, borrow, or steal such scatterers, but it is possible to find ones that scatter equally in two opposite hemispheres of directions (e.g., scattering of sunlight by air molecules).
As with the pile of plates, we define the asymmetry parameter g as the mean cosine of the scattering angle, which has only two values, 1 and -1:
With this definition and the assumption of an isotropic medium the various probabilities in Eqs. (5.40) and (5.41) can be expressed solely in terms of g:
It often is convenient to transform from physical depth z to optical depth t
We have to take care when we encounter this term because total optical depth t is the sum of absorption optical depth Ta and scattering optical depth ts. Authors often write optical depth as t and leave it to readers to guess which one of the three possibilities is meant. If k and 3 are independent of z it follows from Eq. (5.33) that t is physical depth measured in units of total mean free path.
By using Eqs. (5.43) and (5.44) we can write Eqs. (5.40) and (5.41) as where the single-scattering albedo w is ¡3/(13 + k). More compact versions of these equations can be obtained by adding and subtracting them:
In general, optical depth t, single-scattering albedo w, and asymmetry parameter g depend on the frequency of the radiation. The latter two quantities also may vary with physical depth 2 or, equivalently, optical depth; w lies between 0 and 1, g between —1 and 1, although the end points of these two intervals never occur in reality.
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