## Modelling the Earths solar energy balance

The Earth and all its surface ecosystems rely on energy from the sun. Much of this energy for atmospheric motions comes indirectly through radiative, sensible and latent heat transfers from the Earth's surface. As noted above the mixture of greenhouse gases that makes up the Earth's atmosphere is, therefore, central to our understanding of short-term climatic change. This solar derived energy drives the biogeochemical cycles such as the carbon, nitrogen and hydrological cycles. Solar energy also provides one of the vital inputs to the photosynthetic process. Without the inputs of solar energy life as we know it would not exist on planet Earth.

There are numerous models of the atmosphere and many of these are still undergoing further research and development. These models range from the very crude one-dimensional energy models of the 1960s through to the more complex three or four dimensional atmospheric general circulation models (AGCMs) currently used in scientific research and policy discussion. This section describes the science underpinning these models by developing a simple, thermodynamically sound model of the greenhouse effect. More sophisticated versions of modelling global warming are described in the following sub-section.

The solar energy flux just above the Earth's atmosphere is estimated as 1,372 W/m2 — where 1 Watt = 1 Joule/second. This energy flow varies because of solar activity (sun spots), the Earth's distance from the sun and other variables. Nevertheless, despite these changes it is convenient to represent this input of energy as the solar constant:

where Ws is the amount of energy per time period delivered from the sun and R is the radius of the Earth.

In 24 hours the energy is distributed over the entire spheroid of the Earth (assumed to be a sphere) with an area of 4.^.R2. Hence, the average flux of solar energy reaching the Earth is:

and this is the solar flux (Qs). This energy flux covers a large portion of the electromagnetic spectrum.

Not all the energy reaches the Earth's surface by absorption as the Earth also reflects some of the energy back into space. This reflected heat is referred to as the albedo and accounts for 31 per cent of all incoming solar radiation. The remaining 69 per cent of the energy is absorbed by the Earths' atmosphere and is usually re-radiated in the infra-red part of the electromagnetic spectrum.

Under steady-state conditions the Earth is in an energy balance: the amount of incoming solar flux is equal to the amount of energy reflected and re-radiated from the Earth:

Qe = Qs — a.Qs = (1-a).Qs where Qe (W/m2) is the energy flux re-radiated from the Earth; Qs is the incoming solar flux (equation 2.2) 343 W/m2 and a is the albedo of the Earth

Under steady-state conditions we can apply the Stefan-Boltzmann Law of Black-Body radiation. This law states that the energy flux radiation for a black body is a function of the surface temperature of that body raised to the fourth power:

where Q = energy flux (W/m2); \ = Stefan-Boltzmann Constant (5.67 X 10-8 W/m2/K); T = is the temperature in degrees Kelvin (to convert degrees Kelvin (K) to Celsius (C) use the formula K = C + 273.15). By combining equations 2.3 and 2.4 the Earth's average surface temperature is given as:

By substituting the values of a = 0.31 and Qs = 343 W/m2 then equation 2.4 yields a value of: T = 255K (or —18 degrees C).

The average temperature of the Earth would be too cold to support life. Fortunately, the real Earth surface temperature is not so low — it is approximately 33 degrees C higher than the temperature calculated by equation 2.4, or 15 degrees C. The discrepancy between the calculated and actual temperature is due, mainly, but not exclusively, to the presence of greenhouse gases.

Obviously, the model described in this section is very simple. It excludes, for example, any consideration of day and night, nor does it address seasonal changes. Similarly, the model does not incorporate different levels of the atmosphere with different cloud cover. The feedback loops between different surface cover (ice, vegetation and water) and the atmosphere are also omitted. It is possible, however, to develop this argument as a dynamic simulation model by including a set of feedback loops to represent changes in atmospheric CO2.27 It should be noted that this dynamic simulation model excludes any consideration of the dynamics of atmospheric general circulation which is another vitally important aspect of a realistic model of the Earth's changing climate. The model is, therefore, a highly simplified view of the greenhouse effect. Nevertheless, the model does indicate very clearly the role of greenhouse gases in increasing the mean surface temperature of the Earth.

The early energy based models were useful analytical devices but do not treat the world as a three or four dimensional (x, y, z and t (time)) entity and therefore cannot incorporate realistic land surface properties in anything but the crudest of ways. An accurate understanding and reliable prediction of the effect of the greenhouse gases requires consideration of the three dimensional nature of the Earth's surface together with the known dynamics of the atmospheric and oceanic circulation. A fully developed AGCM would incorporate oceanic atmosphere interactions; details of the gaseous transfers between different species of vegetation; details of the albedo effect from dry and wet surfaces (including ice); cloud feedbacks as well as perturbations from volcanic emissions. To describe these missing processes in more detail lead us into a description of the more complex models known as atmospheric general circulation models (AGCMs).

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