The Simplest Climate Model


The Simplest Climate Model, Better Climate Models, and Weather Forecasts End, but Climate Forecasts Continue

The energy from the sun that reaches the top of Earth’s atmosphere is sometimes labeled S, in units such as Watts per square meter (W/m2), and is approximately S=1370 W/m2. Most of the energy leaving the sun misses the Earth and goes streaming off into space, but we intercept a little of it. This total energy reaching the whole Earth is just the Earth’s cross-sectional area multiplied by S, or πr2S, where r is the radius of the Earth.

But, because the Earth is a sphere rotating under the Sun, this energy must be spread around the whole surface area of the planet, including the side facing away from the Sun, with a total area of 4πr2. Hence, the energy available per square meter of Earth’s surface is πr2S/4πr2=S/4.

However, recall that some of this energy is reflected back to space without warming the planet. We call the reflected part the albedo, and for the whole Earth it is roughly 30%, or A=0.3. The absorbed energy is 1-A=0.7. The average energy going to warm the planet is then S(1-A)/4.

The Earth radiates energy back to space, and this can be approximated by “black-body” physics. In this approximation, the outgoing radiation increases with the fourth power of the absolute temperature T (which is how many degrees you are above absolute zero), so outgoing radiation is σT4, where the constant σ, which is often called the Stefan-Boltzmann constant, has a particular numerical value = 5.67* 10-8 W/ m2 /K4 (that is, 5.67 times 10 to the negative eighth power), with temperature in Kelvins (K).(Some people like to write “degrees Kelvin” or “oK”, and the same for “degrees Fahrenheit or oF” or “degrees Celsius or oC”, but it is OK to just use K, F or C.)

Incoming and outgoing energy come into balance, so we have the equation S(1-A)/4=σT4. You can substitute the numbers given just above for S, A, and σ, and then calculate T, the average surface temperature of the Earth. This will give you about 255 K, or -18 C or 0 F, which is well below freezing; the actual average surface temperature is close to 288 K, or 15 C, or 59 F. Our very simple model omitted the greenhouse effect, which keeps the Earth’s average surface temperature above freezing.

Because radiation increases as the fourth power of absolute temperature, the climate is very strongly stabilized. A 1% increase in average temperature causes approximately a 4% increase in radiated power, which means that even a relatively large change in the brightness of the sun, or in other factors affecting the climate, will have a moderately small effect on the temperature. Without this strongly stabilizing effect giving us the climate we have, we might not even be here!

Better Climate Models

Climate models may be the part of the science that most people know the least about. Be very clear-scientists do not tell their computers to produce global warming, and then get excited when global warming comes out of the computer!

The simplest climate model we just discussed shows you a tiny bit of what goes into a real climate model. The starting point is physics. This includes the rules that mass and energy are not created or destroyed but just changed around. The physics also includes interactions between mass and energy-how much energy is needed to evaporate an inch of water per week, for example, or to warm the atmosphere by a degree. Interactions of radiation and greenhouse gases are specified from the fundamental physics worked out by the US Air Force after World War II, and other such studies.

The model also must “know” about the Earth-how much sunshine we get, how big the planet is and how fast it rotates, where the land and oceans are, how much air we have and what it is made of. (Climate models are applied to other planets, and very clearly give different answers because of the differences between the planets.)

All of this information is written down in equations, translated into computer language, and then the computer is turned on. What happens next is remarkable-the computer simulates a climate that looks like the real one. Air rises and rains in the tropics, then sinks and dries over the Sahara and Kalahari. Storms scream out of the west riding the jet stream, and snow grows and shrinks with the seasons across the high-latitude lands.

The model will not be perfect, of course. Suppose you are interested in wind speed. You know from personal experience that you can hide behind a windbreak for relief on a windy day. A forest can serve as a windbreak, giving weaker winds than on a prairie. So, the model must be “told” about the distribution of forests and grasslands (or else must calculate where they grow), and about the “roughness” of the forest and the grass. Scientists have conducted studies on the effects of forests and grasslands on winds, but all studies include some uncertainty. So, the modelers know that the surface roughness in this region must be about this much, but could be a little less or a little more within the range allowed by the data.

The modeler (or more typically, the modeling team) can now “tune” the model. If the winds in the model are a little stronger in some places than in the real world, the modeler may increase the roughness a little, although without going outside the uncertainties. To avoid any biases, different groups in different countries with different funding sources build different models, and tune them in different ways; when all of them agree closely, it is evident that the tuning hasn’t controlled the answer.

Some of the models are used for weather forecasting and for climate studies, and work fine for both. There are differences between weather and climate (see Weather Forecasts End, But Climate Forecasts Continue) - many climate models are simulating changes in vegetation, for example, but if you’re worried about the weather for next week, you don’t really care whether global warming endangers the Amazonian rainforest over the coming decades.

As a general rule, in talking to the public or policymakers, climate modelers rely especially on those results that:

  • are exhibited by a range of models from simple to complex run by different scientific groups;
  • are understood based on the physics;
  • are observed in the history of climate; and
  • are confirmed by recent instrumental observations.