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Thayer Watkins
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 An Extension of the Energy Balance Model of Global Temperature

The energy balance model provides considerable insight into how the gobal temperature of a planet is determined. The energy balance is the equality between the net energy coming in from the Sun and the outgoing thermal radiation. The thermal radiation radiates out from the surface area of the planet (4πR²) but the incoming solar radiation is captured only over the cross-section area (πR²) of the planet, where R is the radius of the planet.

The energy balance model is represented by the equation

#### σT4(4πR²) = (1-a)S(πR²) whichupon division by πR² reduces to 4σT4 = (1-a)S or, equivalently σT4 = (1-a)S/4

where T is global average surface temperature, σ is the Stefan-Boltzmann constant, S is incidence of the solar radiation, and a is the albedo. This energy balance equation can be solved for the global temperature of the planet as a function of the incidence of solar radiation and albedo.

The energy balance equation is said to apply only to "a planet without an atmosphere," but that should instead be "a planet without an atmosphere with greenhouse gases."

If the atmosphere has a component of density ρ which absorbs thermal radiation in a particular frequency band then the proportion absorbed is given by

#### (1 − e-ρz)

where z is a factor which depends upon the depth of the atmosphere and its absorption characteristics.

Let p be the proportion of the thermal radiation that occurs in the absorption band of the greenhouse gas. In general p depends upon temperature but for now treat p as a constant. The total energy radiated per unit surface area is σT4 so the total absorbed per unit surface area is

#### σT4p(1−e-ρz)

and of this one half will be reradiated out into space and one half radiated back down to the planet surface.

An energy balance can be taken either at the planet surface or at the top of the atmosphere. At the surface the balance is between the outgoing energy of σT4 and the net incoming energy of S(1-a)/4+½σT4p(1−e-ρz). (The factor of 4 in the term S(1-a)/4 comes from the ratio of the surface area to the cross-section area of the planet.)

At the top of the atmosphere the balance is between the net incoming solar radiation energy of S(1-a)/4 and the net outgoing energy σT4−½σT4p(1−e-ρz). The equations are

#### σT4 = S(1-a)/4 + ½σT4p(1−e-ρz) (Surface) σT4−½σT4p(1−e-ρz) = S(1-a)/4 (Top of the Atmosphere)

Obviously these two equations are the same.

The solution for the global average surface temperature T is

#### T = [(S(1-a)/4)/(σ(1−½p(1-e-ρz))]1/4

The relationship between temperature T and density ρ is shown below for arbitrary values of the parameters.

## The Proportion of Thermal Radiation in a Given Range

The energy spectrum of the radiation from a body at temperature T is given by

#### (2hν³/c²)/(ehc/kT−1)

where h is Planck's constant, k is Boltzmann's constant, c is the speed of light and ν is frequency. The integral of this over all frequencies is equal to σT4. If the absorption band is from frequency ν1 to frequency ν2 then the energy in this band is

#### ∫ν1ν2(2hν³/c²)/(ehc/kT−1)dν

This is the same as the quantity σT4p. The absorption may be over a set of ranges. A complete treatment must take into account the line broadening of the absorption spectra of the greenhouse gases due to pressure and temperature.

The energy balance equation that determines T is then

#### σT4 − [∫ν1ν2(2hν³/c²)/(ehc/kT−1)dν] (1−e-ρz) = S(1-a)/4

While this equation does not have an analytical solution, its solution by numerical means is not a difficult task.

## Implications Concerning the Linearity of the Trend in Temperatures

The Climatologist Patrick Michaels has promoted a simple and attractive idea that the trend of temperature over time is linear because the response to increasing greenhouse gas concentrations is logarithmic while increase is concentrations is exponential. The convolution of these two functional forms is then linear. A logarithmic response although declining in marginal effect is unlimited whereas the greenhouse gas can do no more than effectively blocking particular channels of the transmission of radiation through the atmosphere. Thus their effect is limited. The effect of even exponentially increasing concentrations of greenhouse gases over time is to asymtotically approach an upper limit. Although the effect may appear to be linear this is only for increases in greenhouse gas densities which are relatively small compared to the saturation level.