## Predictability and nonuniqueness

Assume that the initial state of the climatic system is known, and we want to find a solution to the hydrothermodynamics equations describing the evolution of this system, that is, to predict a change in its state. Here is the question: is such prediction possible and, if so, what is the upper bound for prediction? To answer this question let us remember that the initial state of the climatic system is determined by measurements from an irregular network of stations located at some distance from each other, which means that individual motions with horizontal scales of less than the distance between the stations are not recorded at all. Moreover, initial data are distorted by random errors in measurements, interpolation and rounding-off. Thus even exact solutions of initial hydrothermodynamics equations will entail inevitable errors that increase over time due to the non-linearity of the equations.

The above is aggravated by the fact that the initial equations do not describe the state of the climatic system faultlessly due to the limited knowledge of laws governing the evolution of the climatic system. In addition, as is shown by Lorenz (1975), due to the non-linear interaction of motions of different space and time scales the state of the climatic system can prove to be unstable, in the sense that two states that are initially close to each other will not be so forever. All this results in an increase in discrepancies between predicted and real values.

But any prediction is valid while errors do not exceed the limit of mean climatic dispersions of the predicted values. The period during which this condition is valid is referred to as the limit of predictability (while using deterministic models it is called the limit of deterministic predictability). Thus, the causes of limited predictability of climatic system behaviour are: the instability of the climatic system, inadequate methods of description, and inaccuracy of initial information.

The limit of deterministic predictability is unambiguously connected with the growth rate of errors in initial data or, say, with the time of its doubling. Indeed, with an error of 1 °C in air temperature, a doubling time of three days, an 8 °C mean climatic dispersion of temperature and a constant growth rate of error, the limit of deterministic predictability will be equal to nine days. Such an order of the limit of deterministic predictability is intrinsic to all current operational numerical models of short-range weather forecasting, including the ECMWF operational forecasting system (see Figure 1.2). Data presented in this diagram are interesting in two respects. First, they indicate that the limit of deterministic predictability depends on the quality of the

forecasting model: improvement of the model contributes to an increase in the limit of predictability but only up to a certain point, called the upper limit of deterministic predictability. Second, the limit of deterministic predictability undergoes distinctive seasonal variations, taking maximum values in winter and minimum values in summer. This is associated with the fact that the limit of deterministic predictability is determined by both the growth rate of error and the maximum possible magnitude of error which, by definition, is equal to the mean climatic dispersion of the predicted value (Shukla, 1985).

It is a well-known fact that the growth rate of error is less, and dispersion is much less, in summer in the Northern Hemisphere than in winter. As a result the limit of deterministic predictability is higher in winter than in summer. Also, due to the dominance of the intensity of moist-convective instability in the tropics over the intensity of dynamical (generated by horizontal and vertical wind velocity shears) instability in temperate latitudes, and due to the inadequacy of their parametrization, the growth rate of error in the tropics is higher than that in temperate latitudes, while amplitudes of diurnal oscillations of meteorological characteristics are about the same. Therefore, the limit of deterministic predictability in the tropics is lower than that in temperate latitudes.

The influence of the dispersion magnitude on the duration of the predictability period can be illustrated by two more examples. First: owing to increasing ocean area and decreasing planetary scale disturbances and their variability, the dispersion of meteorological characteristics in the Southern Hemisphere is less than that in the Northern Hemisphere and thus the limit of deterministic predictability in the Southern Hemisphere is less than that in the Northern Hemisphere. Second: the intensity of synoptical disturbances is more than that of planetary disturbances. Hence, all other things being equal (the equal growth rates of error in initial data), the limit of deterministic predictability of synoptical disturbances is less than that of planetary disturbances. This is confirmed by the results of analyses of predictability for 500 hPa geopotential height forecast error fields presented by Tibaldi (1984), according to whom the limit of deterministic predictability for disturbances with wave numbers from 5 to 12 (disturbances of the synoptical scale) is about two weeks, and for disturbances with wave numbers from 0 to 4 (disturbances of the planetary scale) it is about four weeks. Thus, the limit of deterministic predictability increases with the spatial scale of disturbances. This last fact indicates the possibility in principle of forecasting large-scale, long-term oscillations of the atmospheric circulation.

The large-scale, long-term oscillations of the atmospheric circulation can be determined by their own instability and interaction with extremely unstable synoptic motions, or by variations of parameters (particularly of the sea surface temperature, soil humidity and area of snow-ice cover) inducing abnormal distribution of sources and sinks of heat and moisture in the atmosphere and, as a result, changes in amplitude and phase of planetary waves. Accordingly, it is appropriate to speak about the internal and external variability of the climatic system and the corresponding predictabilities of the first and second kind according to Lorenz (1975). By predictability of the first kind is meant the prediction of sequential states of the climatic system (the atmosphere, in this case) at fixed values of external parameters and assigned variations of initial conditions; and by predictability of the second kind is meant the prediction of an asymptotically equilibrium response (of the limiting state) of the climatic system to prescribed changes in external parameters.

It is clear from general considerations that the internal variability of the climatic system has restricted deterministic predictability beyond the boundaries of which the variability serves as the source of an unpredictable noise, while the external variability can be predicted for a longer term, of course, if variations of external parameters are themselves predictable or if they are sufficiently slow. This agrees with the results of numerical experiments by Shukla (1985), bearing witness to the fact that even for monthly mean characteristics of the atmospheric circulation the limit of deterministic predictability is no more than 30 days. Predictability of external parameter anomalies differs: under appropriate conditions the limit of deterministic predictability can be much higher than the value indicated above. It will be recalled that appropriate conditions are those favouring the initiation of strong disturbances of heat sources and sinks in the atmosphere, and the propagation of their influence at considerable distances from the location of a disturbance. In this respect the location of positive sea surface temperature anomalies in the tropical zone is preferable to their location in temperate latitudes because of the abrupt increase in the saturation mixing ratio and, hence, evaporation and latent heat release. As for possible propagation of the influence of anomalies far from their places of localization, this is confirmed by global-scale modes of variability (so-called teleconnections). Quasi-two-year phenomena ENSO serve as good examples of similar kinds of tele-connections (see Section 5.9).

Let us turn back to Figure 1.2 and pay attention to marked fluctuations in the limit of deterministic predictability in the winter time mapping of changes in different types of atmospheric circulation. In particular, the absolute maximum of the limit of deterministic predictability in February 1982 is connected with double (in the Atlantic and Pacific Oceans) blocking of zonal flow, that is, with the formation in the high latitudes of the Atlantic and Pacific Oceans of stationary (for about a week) centres of high atmospheric pressure that prevent the normal propagation of cyclones from the west to the east. This poses the obvious question as to whether this specific case is a reflection of general regularity - the existence of several asymptotic equilibrium states of the atmospheric circulation - and if so, what this means in respect of the predictability of long-period oscillations of the climatic system.

Before answering this question let us recall that a dynamic system is termed ergodic if the equations describing its evolution at random initial conditions and fixed external parameters have a unique possible stationary solution. If the dynamic system is not ergodic, then its behaviour over an infinitely large time interval will depend on the initial conditions. As applied to the climatic system this is equivalent to the fact that external parameters uniquely determine climate in the first case and non-uniquely in the second case.

The idea of the non-uniqueness of Earth's climate was first put forward by Lorenz (1979), who termed ergodic systems transitive, and those systems which do not have the property of transitivity intransitive. The real climatic system, according to Lorenz, is almost intransitive, that is, it shows signs of transitivity and intransitivity simultaneously. Alternation of glacial and interglacial epochs over the last 3.5 million years of Earth's history testifies to this.

It should be borne in mind that the stationary solution discussed above does not necessarily have to be stable in the sense that the system in question approaches or stays for an infinitely long period in the vicinity of some stationary state oscillating around it. If the stationary state is asymptotically unstable then under the influence of external forces the system begins to move away from it and approach another stationary state, which can also be asymptotically stable or unstable. A vertical rod oscillating in the gravity field around the horizontal axis of rotation serves as the simplest example of systems with asymptotically stable and unstable stationary states. A stable state occurs when the centre of gravity is lower than the axis of rotation, and an unstable state occurs when it is higher than the axis of rotation.

Another example is the motion of a liquid in a one-dimensional narrow canal described by the so-called equation of advection:

where u is the current velocity along the x axis.

Let the canal length be L and let it be limited by impermeable walls on both sides where the current velocity vanishes, that is, du/dt + u du/dx = 0,

In accordance with Wiin Nielsen (1975), we present the solution of Equations (1.3.1) and (1.3.2) as the Fourier series

u = Yj "«(0 sin(nfcx), where k = n/L. Substitution of this expression into (1.3.1) yields the following set of ordinary differential equations for coefficients u„ of the Fourier series:

Let us define non-dimensional velocity "t~n and non-dimensional time x using the formulae Vn = m„/n/(2£0), t = ^/(2E0)kt, where 2E0 = Y.n = 1 "«(0 is twice the kinetic energy; the subscript '0' indicates the initial time instant. Then Equation (1.3.3) can be written as drV n 00 1

We consider the two-mode (n = 1,2) system. In this case, according to (1.3.4), we have drjd-c = r^/l, dr2/dr =-rj/2, (1.3.5)

from which it follows that d£/dt = 0; here 2E = (ir\ + ir§), or, taking into account the fact that 2E = 1 at t = 0,

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