Fig 23.14.

Radiative transport in the troposphere.We are considering material a height z above the ground, and the top of the troposphere is at a height zt.

producing gases that will enhance the greenhouse effect. If there is an increase in these gases, then it is possible that the Earth's temperature will go into steady increase. This phenomenon is called global warming. The problem with measuring the effects of global warming is that the increase in any one year is small, and fluctuations due to variations in weather and climate cycles are much larger. However, atmospheric scientists are searching for steady trends (Fig. 23.13b).

The lower part of the atmosphere, heated by radiation from the ground, is called the troposphere. It is that part to which our daily life is confined. We now look at radiative energy transport from the ground into the troposphere, as outlined in Fig. 23.14. The ground is at a temperature Te. We assume that the troposphere has a height zt, and we are interested in energy reaching and leaving some intermediate height z.

The power per unit area reaching z from the ground is aT^. The power per unit area radiated at z is aT4, where T is the temperature at z. However, some of the radiation emitted at z does not escape. It is absorbed at a higher altitude, to be reemitted. The energy density in the layer between z and zt, due to the energy emitted at z is 4aT4/c. Therefore, the energy per unit area in this layer is the energy per unit volume, multiplied by the thickness, zt — z, so energy/area = (4aT4/c)(zt — z)

If all of this energy left z in a time t, the power per unit area is the energy per unit area, divided by t, power/area = (4aT4/ct)(zt — z)

Each time a photon is absorbed and re-emitted, its direction is changed in a random way. We say the photon does a random walk. If the walk is not random, all of the steps are in the same direction. If there are N steps of length L, the distance from the original point is NL. However, in a random walk, a lot of time is spent backtracking. It can be shown (see Problem 23.10) that for a random walk in one dimension (steps back and forth along a line), the distance from the origin after N steps of length L is N1/2L, and for a three-dimensional walk the distance is (N/3)1/2L. (In each case, the total distance traveled by the photons is NL. In the random walk, some of this is lost in the doubling back.)

Using this, the number of steps required for a photon to do a random walk of length zt — z is

where L is the photon mean free path. The length of each step is L, so the time for this number of steps is t = NL/c

Using this, the power per area leaving z becomes power 4a T4 (zt — z) area c(3L/c )(zt — z)2

However, the path length zt — z, divided by the photon mean free path L, is the optical depth t (as discussed in Chapter 6), so power 4 a T 4 area 3 t

Equating this to the power received from the ground aT|, and solving for T, we have

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