Learning from a simple model

The basic case is set up like so: Solar radiation coming in is S=(1-a) \mbox{TSI}/4, where a is the albedo, TSI the solar ‘constant’ and the factor 4 deals with the geometry (the ratio of the area of the disk to the area of the sphere). The surface emission is G=\sigma T_{s}^{4} where \sigma is the Stefan-Boltzmann constant, and  T_s is the surface temperature and the atmospheric radiative flux is written \lambda A=\lambda \sigma T_{a}^{4}, where \lambda is the emissivity – effectively the strength of the greenhouse effect. Note that this is just going to be a qualitative description and can’t be used to quantitatively estimate the real world values.

There are three equations that define this system – the energy balance at the surface, in the atmosphere and for the planet as a whole (only two of which are independent). We can write the equations in terms of the energy fluxes (instead of the temperatures) since it makes the algebra a little clearer.

Surface:  S + \lambda A = G

Atmosphere: \lambda G = 2 \lambda A

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