where Rq is the radius of the Sun, of effective emitting temperature Tq and ro its mean distance from the Earth. The total energy radiated from the Earth, of radius R® and albedo a, is

from which we find that the effective emitting temperature of the Earth, T®, is

From this it is found that T® is about 255 K, and that a 1% change in solar constant corresponds to a change in the radiometric temperature of the Earth of about 0.65 K. Since the emission from the Earth to space is predominantly from the troposphere, with a lapse rate that is at least approximately constant with height, this will be close to the change in the surface temperature to be expected for a 1% change in solar luminosity.

In fact, observations of the total output of the Sun, made with accurate radio-metric instruments orbiting above the atmosphere (§ 10.4), show that the solar constant has varied by less than 0.1% during the last three decades. Most of even that small fluctuation is caused by periodic variations in sunspot number that are so rapid as to have no likely effect on the climate. The remaining, longer-term fluctuations, including those associated with the Sun's 11-year cycle, are predicted by this model to affect the Earth's surface temperature by < 0.01 K.

The first generation of advanced climate models emphasized gradual change, for example the steady increase in global mean temperature that results from the buildup of greenhouse gases, ameliorated by a simultaneous buildup in sulphate and other kinds of aerosol, and possible changes in the amount and type of cloud cover. A more recent goal is the modelling and prediction of abrupt climate change, in which large changes may occur in only a few decades. The challenge is to understand what the stable states of the climate are, how stable the current state is, what transitions are possible, and how and when they are likely to occur.

This new focus has come about because; (a) definite evidence has emerged that rapid climate change, such as the Younger Dryas event, actually occurred in the relatively recent past (around 10000 years ago), and (b) modelling of complex systems in general suggests that the climate system may have multiple equilibria or 'eigenstates' that are stable, separated by unstable or metastable versions that prevail, if at all, relatively briefly. The implication is that climate change may involve gradual evolution for a while, but, once the limits of a stable state are reached, it will then make a rapid transition to the next stable state, which may be quite different. Examples of this kind of climate 'flip' might be from a mainly ice-free planet to one that is heavily glaciated, or a gradual warming of a couple of degrees over a century, say, followed by a much faster increase to a Venus-like state (the 'runaway greenhouse' scenario).

Two simple models, one radiative and the other dynamical, both of which represent some aspects of the real climate system, will be considered to show how multiple equilibria can occur, even in uncomplicated systems. They arise basically because of non-linearities in the dependences between variables, analogous to the behaviour of quadratic and higher-order equations, which have multiple solutions. This will become clear from the following examples.

A simple energy-balance model can also illustrate the concept of multiple climate equilibria, with the implication that rapid climate change can occur between these states. If we now consider the situation where the mean temperature T of the Earth is changing with time t, and the albedo a is a function of T, the balance equation becomes

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