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3.5 The equation of radiation transfer

The equation of radiation transfer describes the modification of the radiation field as it traverses an atmosphere. It has terms related to absorption, emission and scattering of photons by the atmospheric molecules, but can include surface effects. Planetary atmospheres can, as a first approximation, be considered as plane-parallel or simple layers. It is better to simplify and solve the transfer equation to deal with specific problems rather then attempt to formulate a very general solution in three dimensions. We thus focus on the classical solutions for specific cases that were developed by the early workers in the field. These solutions serve as a basis for the development of more complicated methods (see Chapter 6) and for an initial understanding of the key radiation processes in planetary atmospheres.

3.5.1 General form

The general form of the equation of radiation transfer for the radiance, I\ (r, n, t), at a point r and time t, within an atmosphere without significant macroscopic motions, is

---Qt--1--gj- = ri\(r,n,t) - xx(r,t)h(r, n,t), (3.71)

where c is the speed of light, l is the distance travelled by a photon of wavelength A and direction n. Each term has the units of the volume emission, n, watts per cubic meter per steradian per wavelength interval, and x is the extinction coefficient. The first term on the left-hand side represents the time variation of the radiance, while the second term its variation along the photon direction. The first term on the right-hand side represents the volume emission in the direction of the incoming photon, while the second represents the photons that are removed from the incoming direction due to absorption and scattering.

The displacement, dl, is related to a change in position, dr, by the equality ndl = dr. In Cartesian co-ordinates the radius vector can be written as r = xi + yj + zk, (3.72)

and so the direction of the photon can be written as dx dy dz n=dT + i>+di* ^ (3'73)

in terms of the direction cosines between each Cartesian axis and the photon direction. Thus, the equation of transfer can be rewritten as

If^A^YM) + ^ -VI\(r,h,t) =r/\(r,h,t) - x\(r,t)I\(r, h,t), (3.75)

where the nabla operator, V, is given by

dx dy dz

3.5.2 Plane-parallel atmosphere

For a one-dimensional plane-parallel atmosphere, and neglecting temporal and horizontal variations of the radiance, the equation of transfer simplifies to

where ¡j, = cos 0zl and z defines the outward vertical direction of the atmosphere. If the volume emission, n, is a known function of z, then the equation is a simple differential equation that can be solved for each direction j and height z in the atmosphere. If the volume emission involves scattering, as in planetary atmospheres, then the above equation is an integrodifferential equation, since z = 0

FIG. 3.6. Atmospheric layer of optical thickness t and physical thickness zmax. The upwelling direction corresponds to ^ > 0, where p, = cos 6.

the volume emission, n, depends on the radiance, I. If we divide the transfer equation by the volume extinction coefficient, xX, then we obtain it in terms of the source function and optical depth dh{T\,n) T I \ n / \ a4-~-= ja(t"a, m) - b\{tx,i-i).

To solve the above equation we need to specify two boundary conditions, one at the top of the atmosphere (TOA) and one at the base.

3.5.3 Solution for pure absorption

A planetary atmosphere does not emit visible or ultraviolet radiation given its range of temperatures (see Table 3.1). If we neglect scattering, we can solve the transfer equation to obtain its simplest solution, the absorption exponential decay law of Lambert. In this case, the volume emission, nX is zero and so the source function is zero, as is the scattering coefficient, ax. If we consider a slab or single-layer atmosphere without surface reflection, then we solve

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