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An n-Layer
Radiative Transfer Model
of the Atmosphere

This is a presentation and analysis of a radiative transfer model of the atmosphere which involves n isothermal layers. Short wave radiation enters the atmosphere and is transmitted to the absorbing terrestrial layer. The heated terrestrial radiates long wave radiation into the atmosphere where some of its energy is absorbed and some transmitted. An atmospheric layer radiates energy, according to its temperature, upwards and downwards. This radiation is transmitted to the other layers where it may be partially absorbed.

The intensities of the radiation at each level are related by transmissivity matrices, one for upward and another for downward transmissions. Let U is the column vector of upwardly directed energy radiation intensities and D the corresponding column vector of downwardly directed radiation intensities. Let SU and SD be the matrices of the transmission coefficients, where sU,ij is the transmission coefficient to level i from level j. By definition, sU,ij=0 for i≤j. Therefore SU is a lower triangular matrix with zeroes on and above the diagonal. Likewise the matrix SD is an upper triangular with zeroes on and below the diagonal.

There is a small awkwardness concerning the definitions of the vectors. The RU vectors and RD represent radiation from the upper and lower surfaces of a layer whereas U and D represent the vectors of radiation that passes through the midpoints of the layers. This is why the RU element for the i-th layer does not contribute to the i-th element of the U vector. The more exasperating awkwardness is that the natural inclination is to number the layers starting with 1 at the surface layer, but in the vectors the first component is at the top and the numbering goes downward. It is thus very easy to confuse the direction in the physical model and the direction in the matrices.

The transmissivity coefficients are symmetric; i.e., sU,ij=sD,ji. Let RU and RD be the column vectors of upward and downward radiation intensities that are determined by factors other than transmission within the atmosphere. For example, the elements of RU and RD could all be σεiTi4, where Ti and εi are the temperature and emissivity of the i-th level, respectively, and σ is the Stefan-Boltzmann coefficient. Generally, RU and RD are equal except for the elements of boundary layers of the atmosphere. The vectors U and D then satisfy the relations

U=SURU
D=SDRD

The vector of total intensities, level by level, is given by

E = U + D

The absorption of energy at each level is proportional to the radiation intensity at that level. The value of the absorption coefficient at each level is related to the transmissivity coefficient at that level but that relationship need not be made explicit at this point. Let the absorption coefficients be represented in terms of a diagonal matrix with the diagonal elements of that matrix being the absorption coefficients, which of course, by Kirchhoff's Law are equal to the emissivities.

The vector of absorbed energies are given by a column vector equal to AE+F, where F is the column vector of absorbed short wave length radiation. The outputs of radiation, upwards and downwards are given by RU and RD, respectively. The net energy input level-by-level is given by:

−RU −RD + AE + F

The condition for equilibrium is then

−RU −RD + AE + F = 0
or
−RU −RD + A(SURU + SDRD) + F = 0
or (I−ASU)RU + (I−ASD)RD = F

Therefore, if RU=RD=R

R = [2I − ASU − ASD]-1F
or, equivalently
R = {[I − ASU] + [I − ASD]}-1F

Since Ri=σεiTi4, the above relationship immediately gives the temperature level-by-level.


Illustration of the Model

Assume the number of layer is equal to four and the transmissivity of each layer is 0.8. Then the two transmissivity matrices are:

The next step is to multiply these matrices by A and subtract the results from the 44 identity matrix; i.e., The absorptivity/emissivity is 0.2 for all layers so the matrix A has 0.2 on all diagonal elements. Multiplying a matrix by A in this case is equivalent to multiplying each element of that matrix by 0.2. The matrices sought are:

The inverse of the sum of these two matrices is:

Therefore if the energy from short wave length radiation is absorbed only at the surface so that the F vector is, say,

Then the equilibrium energy radiation flux vector R is

A More Complete Analysis

The above example presumes the surface layer is, like the other layers, radiating energy down from its lower layer. When this is not the case the first element of RD is zero. Thus, RU=IR=R and RD=I'R, where I' is square matrix in which the last n-1 rows and n-1 columns constitute an (n-1)×(n-1) identity matrix and the first row and column consists of all zeroes. Thus the equation satisfied for thermal equilibrium are:

R = {[I − ASU] + [I − ASD]I'}-1F

The matrix [I − ASD]I' is

The inverse of the sum of [I − ASU] and [I − ASD]I' is:

For an absorption of 240 W/m² at the surface the equilibrium distribution of energy flux would be:

There is some difference as a result of the modification, but not as large as perhaps might have been expected. The example is still empirically unrealistic in that the transmissivity of the surfacte layer was presumed to be the same as that of the atmospheric layers. The purpose of the example was only to illustrate the computational scheme.

(To be continued.)

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