Equilibrium and stability

If the solar output was to suddenly double, the various components of the climate system would all start to respond in line with their response times. Eventually a new energy balance would be reached and the climate would be in equilibrium. Chapter 1 introduced equation (1.9) to calculate the temperature at the top of the atmosphere; the energy balance equation.

That equation was solved to find T by using present-day values of SQ and a. If the solar output doubled you would obviously get a new temperature. This would be the new equilibrium point, the new balance between incoming and outgoing energy, when the climate system has responded fully to the forcing. As reaching a new equilibrium is not instantaneous, the climate is said to be in transition. This transient climate will represent part of the equilibrium response and the temperature at any time is said to be the realized temperature change. In reality, the nature of the forcing is unlikely to occur instantaneously. Much more likely is a gradual change over time in solar output. Similarly with global warming, as will be shown in Chapter 5, there has been a gradual increase in anthropogenic greenhouse gases. The forcing is gradually changing and the climate's response to that change is also gradual. It may take the oceans hundreds of years to respond to a change in forcings. Therefore even if all the greenhouse gas emissions were to stop tomorrow, the climate would still be committed to a change. This would be in line with the degree of change that has already occurred in the forcing.

Furthermore, the climate system is full of occurrences of feedback. There is even one in the simple equation above, that is albedo and the ice-albedo feedback. Albedo is not a constant but is temperature dependent as described in the previous section. Therefore in the above equation albedo can be rewritten as a function of mean global surface temperature (Ts). When the mean surface temperature is high enough there will be no snow or ice and the albedo will be independent of temperature. The albedo can then be assigned a relatively low but constant value. Similarly, when the planet is cold enough it will become covered with snow and ice and again the albedo will be independent of the surface temperature. This time the albedo will take a high constant value. In between these two extremes the planet will be partially ice covered, with the amount of ice cover, and thus albedo, depending on Ts. Then a simple empirically based equation can be derived between albedo and Ts. A simple equation for outgoing radiation and Ts can also be obtained from observations. Now both the outgoing and incoming radiation can be calculated for various values of Ts (see Kiehl, 1992; note in this book the value for the solar constant used is 1368 Wm 2). The results from these are plotted on a graph in Figure 2.11. The points on the graph where the two lines cross are equilibrium positions when the outgoing radiation balances the incoming radiation. There are three equilibrium points: (1) completely glaciated;

Figure 2.11 Solutions to the incoming solar radiation (S) and outgoing long-wave radiation (F) plotted against temperature revealing three equilibria

(2) partial glaciation; and (3) ice free. Therefore this simple model indicates that the climate system may have more than one equilibrium position. That is it may be possible for the climate system to exist in any one of these states.

But are all these states stable; that is will they persist for long periods of time? To answer this imagine a cone placed on its side as in Figure 2.12b. If you push it then it may roll around but it will stay on its side. This position is said to be neutrally stable. If you place it on its base (Figure 2.12a) then, although a very hard push may knock it over, the cone reacts to any force by returning to its original position on its base. If you balance a cone on its pointed end (Figure 2.12c) then even a slight force will cause it to fall over on to its side. All these positions are equilibrium points but some are more

Figure 2.12 Examples of (a) stable, (b) neutrally stable and (c) unstable conditions

Examples System Stability Condition

Figure 2.13 Concepts of (a) transitive, (b) intransitive and (c) almost intransitive climate system

Source: Peixoto and Oort (1992)

Figure 2.13 Concepts of (a) transitive, (b) intransitive and (c) almost intransitive climate system

Source: Peixoto and Oort (1992)

stable than others. In the case of the cone, it is most stable on its side and least stable on its tip. Figure 2.13 illustrates possible behaviours of the climate system. If there are two different starting points (initial conditions) but the system always evolves to one state, B, then the system is said to be transitive. Any perturbations will eventually lead back to B, so B is a stable equilibrium point. If, however, the system can evolve to different states from the same initial condition, then the system is called intransitive. An almost intransitive system from initial conditions alternates between different resultant states. Many non-linear systems do this and many of the equations that govern the climate system are non-linear. Glacial/interglacial periods maybe a sign of an almost intransitive climate (Peixoto and Oort, 1992). There is, as yet, insufficient data to determine whether the climate is transitive, intransitive or almost intransitive. The latter would be particularly difficult to model. In addition, if the climate is almost intransitive this almost-intransivity could be interpreted by humans as a climate change. Consideration of these different climatic states leads to the question of whether the climate is predictable at all and what role chaos theory may play in determining the climate (Lorenz, 1968).

Chaos theory is particularly concerned with non-linear systems. Rather than define a non-linear equation it is easier to describe a linear equation. A linear equation is one which allows you to draw a straight line. There is a one-to-one relationship between the X and Y values on a graph. Trying to draw the curve from a non-linear equation would result in a much more complex pattern. A simple system that is ruled by a non-linear deterministic equation is the pendulum. You can start the pendulum off from an exact known position at an exact time and from the equations predict, or determine, where it will be in 3 seconds, 4 minutes, etc. As stated earlier, the equations governing the climate system form a non-linear deterministic system and it was thought that these systems were exactly predictable. However, the branch of mathematics called chaos theory is based on the fact that such systems are not necessarily predictable. It turns out that even the simple example, of the pendulum, is extremely sensitive to the initial conditions (Baker and Gollub, 1990). If you set a pendulum swinging from an initial state x and then from x + e, where e is a very small quantity, their dynamical states will diverge quickly, the separation increasing exponentially Very small differences in initial conditions leads to large differences in the final result. 'Prediction becomes impossible' (Poincaré, 1913). From this statement Baker and Gollub (1990) conclude that a chaotic system must resemble that of a stochastic system. That is a system that is subject to random forces. In a chaotic system, however, the irregularities are internal features of the system, not external.

As the equations that govern the climate system form a deterministic nonlinear system, given the initial conditions, it should be possible to solve the equations for some time later. Determining the initial state of the climate, however, would be an impossibility. This leads to the uncomfortable conclusion that weather and climate may never be predictable. It has been suggested that even the movement of a butterfly's wing will cause major changes in the weather end system. Some parts of the climate system may well be chaotic on time-scales of a century to a millennium. Weather systems, such as cyclonic activity, in both the atmosphere and the ocean are undoubtedly chaotic. In the atmosphere the large horizontal scale of these systems make it necessary to consider the whole atmospheric motions as chaotic. There are, however, other features which are stable such as the seasonal cycle in temperature distribution over the continents, monsoons and storm tracks. It is the stability of these features which lead many climatologists to conclude that the response of the climate system to any external forcing will be stable (Cubasch and Cess, 1990). The stable behaviour of the climate system means that we can look for changes by taking time averages. In order to confidently attribute any changes to an external forcing, however, the time averages have to be long compared to any chaotic behaviour.

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