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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 T_{s} and T_{a} 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 T_{s} is the surface temperatre
the emission from the surface is σT_{s}^{4}. The surface also receives
from the atmosphere energy per unit area of
εσT^{4}

The atmosphere receives energy from the incoming shortwave radiation of αF and
absorbs from the radiating surface energy of εσT_{s}^{4}.
It
radiates to the surface and to outerspace energy equal to, in total,
2εσT_{s}^{4}.

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 - 2εσT_{a}^{4} + εσT_{s}^{4} = 0

Surface: (1-α)F + εσT_{a}^{4}
− σT_{s}^{4} = 0

This is a system of two equations in two unknowns, T_{s}^{4} and T_{a}^{4}.
Written in matric form this system is:

Radiative Balance Equation

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^{-1}A(F/σ).

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