2.1.6 Heat balance at the surface

As discussed in section 2.1.1, the incoming solar radiation on a horizontal surface at the top of the atmosphere is about 342 Wm^-2, with roughly 30% of this being reflected back into space. An analysis of the Earth's global heat balance (Fig. 2.19) shows that more that 70% of the reflection takes place in the atmosphere, mainly because of the presence of clouds and aerosols. The remaining 30% is reflected by the surface. By contrast, the majority of the absorption of solar radiation occurs at the surface, which absorbs 2.5 times more solar energy than the whole atmosphere. This shows clearly that the majority of atmospheric warming occurs from below, and not by direct absorption of solar radiation. This important property of the system explains the major characteristics of the Earth's atmosphere, including the vertical temperature profile and the large scale circulation of the atmosphere (see sections 1.2.1 et 1.2.2).

The outgoing longwave radiation required to balance the Earth's Budget at the top of the atmosphere is mainly emitted by the atmosphere and clouds. Among the 396 Wm^-2 emitted by the surface, only 40 Wm^-2 can exit the climate system directly. The large majority of surface longwave radiation is absorbed by the atmospheric greenhouse gases and re-emitted towards the surface where the downward longwave radiation flux (333 Wm^-2) becomes the largest term in the surface heat balance.

In addition to the radiative fluxes, the surface and the atmosphere exchange heat through direct contact between the surface and the air (sensible heat flux or thermals) as well as through evaporation and transpiration. Indeed, when evaporation (or sublimation) takes place at the surface, the latent heat required for the phase transition is taken out of the surface and results of a surface cooling. Later, mainly during the formation of clouds, the water vapour condensates and the latent heat is released into the atmosphere. This leads to a net mass and heat transfer from the surface into the atmosphere, which is one of the main drivers of the general atmospheric circulation.

The fluxes of sensible and latent heat are generally estimated as a function of the wind speed at a reference level, and the difference in temperature (for the sensible heat flux F_SH) or specific humidity (for the latent heat flux F_LH) between the surface and the air at this reference level, using classical bulk aerodynamic formulae:

where U_a, T_a, q_a are the wind velocity, air temperature and specific humidity at the reference level (generally 2 m or 10 m), T_s and q_s are the surface temperature and specific humidity at the surface, c_h and c_L are the aerodynamic (bulk) coefficient. In general, they are function of the stability of the atmospheric boundary layer, the roughness of the surface, the wind speed and the reference height. In the majority of cases, c_h and c_L are not too different from each other and their value ranges from 1.10^-3 to 5.10^-5. The highest values occur with unstable boundary layers and very rough surfaces which tend to generate strong turbulent motions and thus higher exchanges between the surface and the air than quieter situations.

The specific humidity, q_s, above a wet surface is generally very close to saturation. It can thus be expressed using the Clausius-Clapeyron equation, which shows that the amount of water vapour in the air at saturation is strongly dependent on the temperature. For instance, the amount of water vapour that can be present in the atmosphere at a temperature of 20°C is more than three times higher than at 0°C. As a consequence, the evaporation and the latent heat flux are much larger at low latitudes than at high ones. The latent heat flux is thus larger than the sensible heat flux at low latitudes, while the two fluxes are generally of the same order of magnitude over the ocean at high latitudes. The ratio between the sensible heat and latent heat fluxes is usually expressed as the Bowen Ratio B_o:

Over land surfaces, the latent heat flux is a function of the water availability and B_o can be much higher than unity over dry areas.

The heat balance shown in Figure 2.19 for the whole Earth can also be computed for any particular surface on Earth. This is generally the method used to obtain T_s. Let us consider a unit volume at the Earth's surface with an area of 1 m² and a thickness h_su (Fig. 2.20). h_su is supposed to be sufficiently small to safely make the approximation that the temperature is constant over h_su and equal to T_s. The heat balance of this volume can then be expressed as:

The left-hand side of the Eq. 2.36 represents the heat storage in the layer h_su (see section 2.1.5.1). F_sol is the incoming solar flux at the surface which is a function of the incoming solar radiation at the top of the atmosphere and of the transmissivity of the atmosphere (related to the presence of clouds, aerosols, the humidity of the air, etc). A fraction $\alpha$ of F_sol is reflected by the surface and not absorbed. F_{IR$\downarrow$} is the downward longwave radiation flux at surface. This flux is caused by the emission of infra-red radiation at various levels in the atmosphere. It is thus a complex function of the temperature and humidity profiles in the atmosphere, the cloud cover and the height of the clouds, the presence of various greenhouse gases (in addition to water vapour), etc. The longwave upward radiation flux F_IR$\uparrow$ can be computed using the Stefan-Boltzman law while the expressions for F_SE and F_LE are given by Eq. 2.33 and 2.34. F_cond, the flux from below the surface, is a conduction flux for solid surfaces (such as the ground and the ice) that can be represented following the Fourier's law. For the ocean, this flux is related to the dynamics of the oceanic mixed layer. Additionally, if the media at the surface is (partly) transparent, a fraction of the radiation is not absorbed in the layer of thickness h_su and must be subtracted from the term ( 1 - $\alpha$ F_sol) in Eq. 2.36. For the other fluxes, the exchanges take place in a very shallow layer and can reasonably be considered as purely surface processes.

Figure 2.20 displays a relatively simple situation where the surface (i.e. the interface between the atmosphere and the material below) is clearly defined. In complex terrain with very rough topography, for instance over forests or urban areas, defining the lower limit of the atmosphere is less straightforward. Computing the surface fluxes in these regions is a very complex issue which is currently the subject of intense research.

When snow or ice is present at the surface, the temperature T_s cannot be higher than the freezing point of water. As a consequence, Eq. 2.36 remains valid as long as T_s is below the freezing point. When surface melting occurs (i.e., when T_s equals the freezing point of water) an additional term, corresponding to the latent heat of fusion required to keep the temperature unchanged, must be added to the right-hand side of Eq. 2.36.