## The Energybalance Equation

So far we have given separate consideration to three major determinants of the climatic environment: (i) the net flux of the radiation energy from the Sun to the ground (Section 2.8), (ii) its subsequent use in heating both the air (Sections 1.6 and 3.2) and the ground (Section 3.5), and (iii) the evaporation of water (Section 4.1). In later chapters, we shall elaborate on the ways in which these account for the humidity, cloudiness, rainfall and wind, which—along with temperature—constitute the weather in the short run and the climate in the long term. But at this point it is worth while to tie together what has been said about radiation, heating, and evaporation, as well as the melting of ice, photosynthesis, artificial heating (Section 3.7), wind friction and horizontal advection.

The processes just mentioned are all equivalent to flows of energy which can be combined by means of an energy balance, a special case of the general balance equation. In brief, this states that change equals the difference between input and output. We have already considered another special case, that of the water balance of an area (Section 4.5), which states the obvious truth that the change of water level over a specified period equals the input minus the output during that time. Here, instead of water, we balance the amounts of energy flowing in unit time:

of energy of energy stored energy or stored energy of energy of energy

Either of the above is the energy-balance equation. An energy balance can be considered for any volume (in terms of joules per second, i.e. watts) or any surface (where the fluxes have units of W/m2). The volume might be a person's body (Section 5.5), the bulb of a thermometer (Chapter 6), a parcel of air (Chapter 7) or the global atmosphere (Chapter 12). On the other hand, the surface that we consider most is that of the ground.

The energy-balance equation applies either to instantaneous conditions or to fluxes averaged over some period, such as a day or year. The equation is universally true, expressing what is called the 'principle of the conservation of energy' or the 'first law of thermodynamics' (Chapter 7).

Surface Energy Balance

The equivalent to the previous equations is the following, for energy fluxes at the ground surface:

W/m2

where Rn is the input of energy in the form of net-radiation, L the latent heat of evaporation (J/kg, Section 4.1), E the rate of evaporation (kg/m2.s), H the convective flux of sensible heat away from the surface (Section 3.1), and G the heating of the ground (Section 3.5). The product L.E is the latent-heat flux (Section 4.5), often written as simply LE for short. An equation like that applies to most surfaces. Exceptions include an ocean surface, fast-growing vegetation or melting snow, which would require extra output terms on the right of the equation, for advection, photosynthesis (Section 1.2) or latent-heat-of-melting, respectively.

Notice the difference between the two kinds of equation involving net radiation Rn, here and in Section 2.8, respectively. The expression in Section 2.8 defines the net radiation, gives it a label; Rn is the sum of all radiation fluxes onto the surface. On the other hand, the energy-balance equation here is more than word-play, it describes an important quantitative feature of reality, the conversion of Rn into non-radiative forms of energy.

The simple form of the energy-balance equation at the ground's surface is shown in Figure 5.1. In that case, a net radiation towards the surface is regarded as positive and LE, H and G are positive if they move away from it. LE and H are usually positive, but the LE term is negative if dew forms, since the latent-heat transfer is then towards the surface. Likewise, the sensible-heat flow H is negative if the air is warmer than the surface, so Figure 5.1 The chief components of the energy balance at the ground. Incoming net radiation Rn is balanced by the fluxes away from the surface, in heating the ground G, heating the air by convection H and in evaporation LE.

that heat moves downwards to the surface. Also, the flux of heat conducted through the ground G is usually negative at night, as it then flows upwards towards the cooling surface.

The ground-heat flux G in the equation above is typically about 20 per cent of Rn over an hour or so. However, it is negligible over the course of a whole day or more, since the daytime absorption is almost exactly cancelled by the night-time loss. The error in ignoring G is then no more than the uncertainty in measuring Rn. In that case, Rn is balanced by the sensible-heat flux H and the latent-heat flux LE, together. If the surface is also dry (so that LE is zero), Rn equals H, i.e. all the net-radiation input goes into heating the adjacent air. For example, H in central Australia is shown in Figure 5.2 as 40-60 W/m , which accounts for much of the net radiation, shown in Figure 2.17 as 80 W/m2. On the other hand, H over the subtropical oceans is small compared with Rn, indicating that Rn there is largely balanced by LE, which is large (Figure 4.11).

Applications

A zero heat-storage term G means that temperatures are steady, and the balance Figure 5.2 Annual mean sensible-heat flux H from the Earth's surface to the atmosphere, in units of W/m2.

equation becomes simply the statement that inputs and outputs are equal, as in the case of the entire Earth (Note 5.A). Such an equation of equilibrium allows us to calculate the temperature of a white vehicle, in comparison with a black one in a hot climate (Note 5.B).

An energy-balance equation permits deducing any one of the terms from a knowledge of the others. For instance, consider the case of an ocean surface over a month. The overall storage term G is negligible over such a period, with daytime heat gains cancelled by nocturnal loss. Also, measurements show that there is little difference between the temperatures of the air and water near their interface, which implies zero sensible-heat flux H (Section 3.2). Moreover, advection is negligible in the case of a large ocean basin, where the affected edge is only a small fraction of the entire volume. Therefore, the energy-balance equation reduces to an equality between Rn and LE. So we can obtain the approximate long-term rate of evaporation from an ocean (which is hard to measure) by the relatively straight-forward determination of the net-radiation flux Rn there (cf. Figure 2.17 and Figure 4.11).

Also, the energy balance helps us to understand the processes causing cooling at night (Note 5.C) and to estimate evaporation (Note 5.D). In addition, the energy balance explains a sol-air temperature, the reading of a thermometer exposed to the Sun's radiation and the ambient wind (Note 5.E). It is the temperature reached by any exposed dry surface (Note 5.B) and by people outdoors, so that bright calm days feel acceptably warm even when the air is cold. Further examples of the usefulness of the energy balance are considered in the following sections.

### Other Components of the Energybalance Equation

Relatively small amounts of heat are used in heating of the ground G, in photosynthesis and in frictional heating of the ground by wind. Plants also use merely a small fraction of the incoming solar radiation Rs (Note 2.I). Heating due to the friction of the wind on the ground is trivial compared with net-radiation values (Section 2.8); it is less than 0.2 W/m2, even with the high winds in Antarctica, for instance (Chapter 16). Wind is more important in causing the advection of sensible and latent heat (Notes 3.A and 4.D) than as a source of energy itself.

Until now we have mainly considered vertical heat fluxes, but there are horizontal transfers of energy too. These occur as sensible heat in the wind (especially during storms) and in ocean currents, and as latent heat in moist wind. There is a net advection of heat towards an area (i.e. heat-flux convergence) when the inflow exceeds the outflow, and this has to be included in the energy-balance equation for the area. Such contributions are important on a global scale (Note 5.F, Chapters 11 and 12). Figure 5.3 Long-term global atmosphere, representing longwave shortwave radiation sensible-heat flux and are percentages of the different numbers are other authors.

average fluxes of energy in the with the underlined numbers radiation, while D denotes diffuse S is direct solar radiation, H is L.E latent-heat transfer. Numbers incoming solar radiation. Slightly given for the various fluxes by

Figure 5.3 Long-term global atmosphere, representing longwave shortwave radiation sensible-heat flux and are percentages of the different numbers are other authors.

average fluxes of energy in the with the underlined numbers radiation, while D denotes diffuse S is direct solar radiation, H is L.E latent-heat transfer. Numbers incoming solar radiation. Slightly given for the various fluxes by