Concept of critical levels

Many of the crucial values for society are being changed under climatic variations more or less smoothly: crop yield, water storage, water level of rivers, sea level etc.. (Note that their ``smoothness'' is determined, in the first place, by their measurability.) Therefore, these values can be considered, generally, as continuous functions of climate change, which is a consequence of the emission of ``greenhouse gases''. This process is determined mainly by the structure and organisation of the world energy system. However, a social perception transforms the continuous range of those values to a discrete set, namely ``good-medium-bad-disaster''. For instance, the difference between a bad crop yield and a catastrophic one may be less than the difference between medium and bad yields, but the social consequences of ``bad'' and ``catastrophic'' yields are not comparable. The impossibility of navigation is a result of the fact that the water levels sink below some critical threshold. A local agricultural disaster may certainly be compensated at the regional scale (with the help of market mechanisms), but there may exist a critical level for total crop production which destroys the market as a whole. For many social systems such critical thresholds (either scalars or vectors or surfaces in the space of basic variables) exist. If a system crosses such boundaries then the system's homeostasis is destroyed.

Therefore, we define a critical event as a crossing of the homeostasis boundary for some social system (see below) after which this system cannot be reconstructed (or will not reconstruct itself).

Note that in numerous works the ``CO-doubling'' is considered as a crucial point for climate, biota, etc. So, this is not a critical event, and what is more, the state of the biosphere corresponding to CO-doubling is not unique, because it depends substantially on the path of transition from the present state to the state where CO concentration will be doubled. The concentration of carbon dioxide in the atmosphere is simply one of many strategic variables of the whole system.

Suppose there is a social system, the state of which is described by the vector x (in general case x may depend on time, that is, to be a dynamic variable). We suppose also that the state depends on climate which is described by the vector , so that .

Let be a homeostasis domain for this system. It is obvious that the critical level, , is defined by the equation . In turn, the inequality induces the set of admissible x with the boundary .

Suppose that the inverse mapping does exist. Then we can calculate the set of admissible climates with the boundary , where . Note that the mappings and may be non-unique.

The solution of this criticality problem is trivial: if the predicted climate does not belong to the set then the state of the system is catastrophic.

We illustrate these abstract considerations by the following concrete example. Let our social system be the agriculture system producing cereals. The state of the system is described only by the crop yield, that is, by the scalar x. If then we have an ``agricultural disaster'', the value is determined by economic and social arguments. The homeostasis domain for this system is defined by the inequality . Then , (see Fig. 1). If we assume that the crop yield depends only on the annual temperature , then the climate is described by the scalar and . As a rule, the dependence is uni-modal, and, how we can see in Fig. 1, the mapping is non-unique. Therefore the set of admissible temperatures is the interval with the boundary . These two boundary points correspond to two solutions of the equation .

  
Figure 1: To the definition of the sets and : ; ; ; ; .

Revenons à nos moutons, we can say that the solution of our criticality problem is trivial if we have:

1. an ideal model for the system, that is, for each there are a finite number of separated x (and vice versa). (If, for instance, the dependence of crop yield on the temperature is a probability function, then this assumption is not true. The ideal model is a deterministic one, which predicts the state x corresponding to climate with unit probability, possibly apart from hysteresis.);
2. an ideal climatic prediction (again with the unit probability).

In actuality, we neither have an ideal model nor an ideal forecast.

Let us suppose that there are different deterministic models and that each of them possesses a different ``predictive power'' depending on its structure, complexity, scientific uncertainties etc.. The simple way to describe this power is to associate some specific probability to each predicted value . Thus, some probability measure can be constructed on the basis of the set of models . Another way is to use some stochastic model with a probability measure. In these cases we can formulate the following probabilistic statement: the inequality takes place for any with the specific probability .

Next we have to take into account that there is no ideal prediction for . Scientific uncertainties in climate prediction will allow us to predict the value only with some probability , so that instead of unique prediction for we have a set of values for , and each of them can be realised with a specific probability. Certainly, the suggested method of uncertainty analysis is not unique in a problem of GW, there are other approaches (see, for instance, [4]), which can also be used in calculation of corresponding probabilities.

And, finally, for the given the probability of the event is equal to:
 
under the normalisation condition
 
There is not a problem in generalising this approach to the case when is a functional and the critical event is also a functional, etc..

The probability R can be considered as a measure of the risk for the catastrophic event . But we also have to keep in mind the following facts:

1. The value is defined by social factors only. This means that the event could be either admissible or inadmissible depending on the social consequences.
2. The risk level is defined by politicians, who have to compare two sets: and the corresponding . The choice of the risk level depends on many factors: the state of society, economy, social stress etc..
3. The main task of the scientific community is to provide the politicians with options: to present them the values for and , and illustrate the events with facts and pictures.

Let us now consider the following situation: We have determined the value , the probability predictions for climate change, that is, , and by calculation of we have found out that , where is an admissible level of risk. What kind of action can be started in such a case?

1. We can take steps in order to lower the critical level to such that . In the case of crop production we can, for instance, import an additional amount of grain (adaptive strategy).
2. We can select and cultivate a new variety of plants in the region (that is, we change ).
3. At the global level we can reduce the green-house gases emissions in order to change and, by the same token, until it becomes equal .
4. Finally, we can take the risk and do nothing.

The algorithm for risk assessment consists of the following sequence of steps:

1. By using a model for the social system which is driven by the climatic input , we construct the -domain with the boundary .
2. On the set of possible , we construct the probability measure .
3. Using we calculate the probability for the event (for any ).
4. We determine the probability distribution , i.e., the probability of the realisation of the given climate scenario .
5. Calculating the risk level we construct the table .
6. Politicians choose the pair .
7. For the chosen pair we have to find either the appropriate structure of the system , or the appropriate climate .

Certainly, it is easier said than done (especially this is true for the first two points), but for relatively simple systems it is possible. We have shown above how to do this, for instance, for such a system as an agricultural one.