Climate Model - Zero-dimensional Models

Zero-dimensional Models

A very simple model of the radiative equilibrium of the Earth is

where

• the left hand side represents the incoming energy from the Sun
• the right hand side represents the outgoing energy from the Earth, calculated from the Stefan-Boltzmann law assuming a model-fictive temperature, T, sometimes called the 'equilibrium temperature of the Earth', that is to be found,

and

• S is the solar constant – the incoming solar radiation per unit area—about 1367 W·m−2
• is the Earth's average albedo, measured to be 0.3.
• r is Earth's radius—approximately 6.371×106m
• π is the mathematical constant (3.141...)
• is the Stefan-Boltzmann constant—approximately 5.67×10−8 J·K−4·m−2·s−1
• is the effective emissivity of earth, about 0.612

The constant πr2 can be factored out, giving

Solving for the temperature,

This yields an apparent effective average earth temperature of 288 K (15 °C; 59 °F). This is because the above equation represents the effective radiative temperature of the Earth (including the clouds and atmosphere). The use of effective emissivity and albedo account for the greenhouse effect.

This very simple model is quite instructive, and the only model that could fit on a page. For example, it easily determines the effect on average earth temperature of changes in solar constant or change of albedo or effective earth emissivity.

The average emissivity of the earth is readily estimated from available data. The emissivities of terrestrial surfaces are all in the range of 0.96 to 0.99 (except for some small desert areas which may be as low as 0.7). Clouds, however, which cover about half of the earth’s surface, have an average emissivity of about 0.5 (which must be reduced by the fourth power of the ratio of cloud absolute temperature to average earth absolute temperature) and an average cloud temperature of about 258 K (−15 °C; 5 °F). Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature 285 K (12 °C; 53 °F)).

This simple model readily determines the effect of changes in solar output or change of earth albedo or effective earth emissivity on average earth temperature. It says nothing, however about what might cause these things to change. Zero-dimensional models do not address the temperature distribution on the earth or the factors that move energy about the earth.