Two-stream approximation

Photolysis rates (J values) are calculated on-Une every chemical time step using the method of Landgraf and Crutzen (1998). In this method, radiative transfer calculations are done using a delta-two stream approximation for eight spectral... 

Meador, W.E., and W.R. Weaver, Two-stream approximations to radiative transfer in planetary atmospheres A unified description of existing methods and a new improvement. J Atmos Sci 37, 630, 1980. 

The next step is to relate cloud albedo Rc to xr. The two-stream approximation for the reflectance (albedo) of a nonabsorbing, horizontally homogeneous cloud gives (Lacis and Hansen 1974)... 

We now introduce a fifth assumption critical to determining an analytic solution to Eq. (5) We assume that the radiance is slowly varying with the variable n, so that we can replace the entire angular distribution (at a given level) with two values, and I. This is the gist of the famous two-stream approximation. It allows us to make the following simplifications ... 

From Eq. (7), the irradiance gradient is zero in RE, which implies a constant (net) irradiance in the vertical direction. As discussed previously, the irradiance leaving the atmosphere (at t = 0) is equal to the irradiance at any arbitrary optical depth and is given in the two-stream approximation, Fir) 27Tp[I+ir)—I ir) = 2jrpaBT ir)/jt. Note carefully that for consistency with the two-stream approximation the exact result ias T ) has been replaced by the approximate result i2poB T ). But, also note that if we set p equal to 1/2, the results are identical. This is the value we will adopt here, but we will retain the notation p. 

The two-stream approximation is just such a solution. The continuous radiation field is replaced by two directed beams, one up and one down. Because the solution is analytic, it is helpful to separate the azimuthal and polar angles. We write the phase function [cf. Eq. (2.1.3)] as a finite series of Legendre polynomials ... 

Scattering atmospheres the two-stream approximation we establish the identity... 

In the two-stream approximation the continuous radiation field is replaced by one traveling in only two directions. Hence the integral in Eq. (2.5.10) is replaced by a sum,... 

Certain modiflcations of the line shape and strength occur when scattering is introduced. Figure 4.1.4 illustrates the physical process, and should be contrasted with the nonscattering model shown in Fig. 4.1.1. A useful analytic representation follows from the two-stream approximation developed in Chapter 2. Considering... 

Because only fluxes are physically significant in our problem, the two-stream approximation is deemed adequate. Hence the radiation fields are restricted to the directions fio = l i = —At-i = 1 J i- It can be shown (Samuelson, 1983) that a solution of Eq. (9.1.4) leads to a flux divergence... 

An analytic solution to Eq. (9.1.24) has been obtained by the method of discrete ordinates (Samuelson, 1983), which is a generalization of the two-stream approximation considered in this chapter. The solution is then used to obtain expressions for the flux from Eq. (9.1.18) and the Planck intensity from Eq. (9.1.17). The upper boundary condition... 

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