2.1.1 The heat balance at the top of the atmosphere: a global view

Nearly all the energy entering the climate system comes from the Sun in the form of electromagnetic radiation. Additional sources are present, such as geothermal heating for instance, but their contribution is so small that their influence can safely be neglected. At the top of the Earth's atmosphere, a surface at the mean Earth-Sun distance perpendicular to the rays receives about 1368 W/m² (see also Fig. 5.27) This is often called the Total Solar Irradiance (TSI) or solar constant S₀. A bit less than half of this energy comes in the form of radiation in the visible part of the electromagnetic spectrum, the remaining part being mainly in the near infrared, with a smaller contribution from the ultraviolet part of the spectrum (Fig. 2.1).

On average, the total amount of incoming solar energy outside the Earth's atmosphere (Fig 2.2) is the solar constant times the cross-sectional surface (i.e., the surface that intercepts the solar rays, which corresponds to a surface $\pi$R² where R is the Earth's radius of 6371 km²). For simplicity and because it is a reasonable approximation, we will neglect the thickness of the atmosphere compared to the Earth's radius in our computations of distances or surfaces. Some of this incoming flux is reflected straight back to space by the atmosphere, the clouds and the Earth's surface. The fraction of the radiation that is reflected is called the albedo of the Earth or planetary albedo (${\alpha }_p$). In present-day conditions, it has a value of about 0.3.

In order to achieve a heat balance, the heat flux coming from the Sun must be compensated for by an equivalent heat loss. If this were not true, the Earth's temperature would rapidly rise or fall. At the Earth temperature, following Wien's Law, this is achieved by radiating energy in the infrared part of the electromagnetic spectrum. As the radiations emitted by the Earth have a much longer wavelength than those received from the Sun, they are often termed longwave radiation while those from the Sun are called shortwave radiation. Treating the Earth as a black body, the total amount of energy that is emitted by a 1 m² surface ($A\uparrow$) can be computed by Stefan-Boltzmann's law:

where $\sigma$ is the Stefan Boltzmann constant ($\sigma$=5.67 10^-8 W m^-2 K^-4). This equation defines T_e, the effective emission temperature of the Earth. The Earth emits energy in all directions, so the total amount of energy emitted by the Earth is $A\uparrow$ times the surface of the Earth, 4$\pi$R². To achieve equilibrium, we must thus have (Fig. 2.3) :

This leads to

and finally to

This corresponds to T_e=255K(=-18°C). Note that we can interpret Eq. 2.3 as the mean balance between the emitted terrestrial radiation and the absorbed solar flux for 1 m² of the Earth's surface. As shown above, the factor 1/4 arises from the spherical geometry of the Earth, because only part of the Earth’s surface receives solar radiation directly.

The temperature T_e is not a real temperature that could be measured anywhere on Earth. It is only the black body temperature required to balance the solar energy input. It can also be interpreted as the temperature that would occur on the Earth's surface if it were a perfect black body, there were no atmosphere, and the temperature was the same at every point.