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 The Simplest Climate Model Demostrating the Greenhouse Effect

This is a presentation and analysis of a minimalist model of radiative heating that incorporates the elements of the greenhouse effect. Consider a blackbody plane under a layer of atmosphere. The atmosphere is partially transparent to the incipient shortwave radiation but absorbs a portion of the longwave radiation generated by the heated blackbody plane. The atmospheric layer radiates as a gray body to the surface and to outerspace.

Let Ts and Ta be the absolute temperatures of the surface and the atmosphere, respectively. Let ε be the emissivity and absorbity of the atmosphere. Let α be the proportion of incipient shortwave absorbed by the atmosphere.

If F is the intensity of the incoming shortwave radiation, then the energy per unit area reaching the surface is (1-α)F. If Ts is the surface temperatre the emission from the surface is σTs4. The surface also receives from the atmosphere energy per unit area of εσT4

The atmosphere receives energy from the incoming shortwave radiation of αF and absorbs from the radiating surface energy of εσTs4. It radiates to the surface and to outerspace energy equal to, in total, 2εσTs4.

At equilibrium the net energy flow both to the surface and the atmosphere must be zero so the equations to be satisfied are:

#### Atmosphere: αF - 2εσTa4 + εσTs4 = 0   Surface: (1-α)F + εσTa4 − σTs4 = 0

This is a system of two equations in two unknowns, Ts4 and Ta4. Written in matric form this system is:

Let the matrix on the left above be represented as M, the column vector of the fourth powers of temperatures as S and the column vector on the right as A. The system is then MS=A(F/σ) Then the solution for the column vector of the fourth powers of temperatures S is equal to M-1A(F/σ).

The determinant of M is 2ε−ε², which is (2−ε)ε. The inverse of M is then ) ">

Therefore the solution is

The Solution

(To be continued.) 