*This note refers to almost-intransivity in nonlinear systems applied to climatology. For the more general case, see the definition of almost-intransivity at [[intransivity]].* >[!abstract] >Almost-intransivity refers to a climate system with more than one stable equilibrium and the ability to move from one to the other due to some external perturbation. [[Lorenz, 1968|Lorenz (1968)]] and [[Lorenz, 1976|Lorenz (1976)]] discussed how a temporary environmental change, such as in solar activity, could cause such a switch. [[Gleick, 1987|Gleick (1987)]] further popularized the concept for lay readers. >[!quote] >In the case of nonlinear equations [such as those that govern Earth's climate], the uniqueness of long-term statistics is not assured. From the way in which the problem is formulated, the system of equations, expressed in deterministic form, together with a specified set of initial conditions, determines a time-dependent solution extending indefinitely into the future, and therefore determines a set of long-term statistics. The question remains as to whether such statistics are independent of the choice of initial conditions. We define a *transitive* system of equations as one where this is the case. If, however, there are two or more sets of long-term statistics, each of which has a greater-than-zero probability of resulting from randomly chosen initial conditions, the system is called *intransitive*. >[...] >This leads us to the concept of a special type of transitive system which, for want of a standard mathematical term, I shall call *almost intransitive*. In an almost intransitive system, statistics taken over infinitely long time intervals are independent of initial conditions [like a perfectly intransitive system], but statistics over very long but finite intervals depend very much upon initial condition [like a transitive system]. Alternatively, a particular solution extending over an infinite time interval will possess successive very long periods with markedly different sets of statistics ([[Lorenz, 1968]]). >[!quote] >There are extremely simple and also very complicated systems of equations possessing solutions which behave in one manner for an extended period of time, and then change more or less abruptly to another mode of behavior for an equally long time. Such systems have been described as *almost intransitive* ([[Lorenz, 1970]]). >[!quote] >Both transitive and intransitive systems occur in nature. [...] If the system is truly intransitive, changes from one possible climate to another will never occur on their own accord. We therefore note the possibility that if the system should be intransitive under a constant environment, a temporary change in the environment, perhaps due to some anomalous solar activity, could cause a new climate to develop. If the environment should then return to its original state, one of the alternative climates compatible with this state might become established instead of the original climate ([[Lorenz, 1976]]). >[!quote] >An almost-intransitive system displays one sort of average behavior for a very long time, fluctuating within certain bounds. Then, for no reason whatsoever, it shifts into a different sort of behavior, still fluctuating but producing a different average. [...] Then, to explain large changes in climate, [the people who design computer models] look for external causes — changes in the earth's orbit around the sun, for example. Yet it takes no great imagination for a climatologist to see that almost-intransivity might well explain why the earth's climate has drifted in and out of long Ice Ages at mysterious, irregular intervals. If so, no physical cause need be found for the timing. The Ice Ages may simply be a byproduct of chaos ([[Gleick, 1987]], p. 170). >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —