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Deterministic nonperiodic flow.

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Lorenz, E.N., 1963. Deterministic nonperiodic flow. Journal of Atmospheric Science. vol. 20, 130–141.

Essay about this article

This pathbreaking paper by Lorenz published in 1963 presents a simple system of equations representing cellular convection in which all the solutions are unstable and most are nonperiodic. The implications were immediately obvious for long-range weather prediction, intriguing and significant for climate modeling, and are still being worked out today in many interdisciplinary fields.

There have been many external influences on the atmospheric science including discoveries in other fields, new technologies such as digital computers and satellites, and even social pressures. But Edward Norton Lorenz, a founder of the new science of chaos theory, is the meteorologist whose work has had the most significant impact on the world at large. His computer simulations of the atmosphere showed that the final state of a dynamical system exhibits sensitive dependence on the initial state, and that small alterations in this initial state can result in huge differences in outcome. In popular usage the term “butterfly effect” was attached to this concept, while in scientific circles, the hauntingly beautiful image of his eponymous “Lorenz attractor,” with its elegant butterfly wings, has become an icon of chaos theory. His work implied that climate change could cause catastrophic “surprises” and that models needed somehow to take this non-linear behavior into account.

Lorenz’s work on the unpredictable behavior of complex natural phenomena inspired a wave of interdisciplinary investigation into a new universe of non-linear systems. Because of him, we have begun to understand a broad range of phenomena not as disordered, discontinuous and erratic, but as ultimately beautiful and clearly governed by a new set of concepts—and limitations. Because of his work on strange attractors, we can begin to refer to an earlier era in science as BC: “before chaos.”

In the 1950s Lorenz, realizing that the equations of meteorology were not conservative, developed a simple three-variable model, which he programmed into a Royal McBee LGP-30 desk-top computer. The aperiodic solutions, when displayed graphically, traced out the non-repeating butterfly pattern which is now so widely known in chaos theory. He also discovered, in the course of programming the computer, that small errors of truncation (representative of uncertainties or inaccuracies in observation) quickly amplified so much that they drowned out the signal. This implied that if the atmosphere behaved this way, then long-range forecasting was impossible because measurements can never be completely accurate. Although at first he had to search for a set of equations that produced a strange attractor, he later discovered many more that behaved this way.

Lorenz wrote the following in 1996 in his essay on the evolution of dynamic meteorology. Its implications are relevant to climate modeling as well and are worthy of extended quotation. A chaotic system is one that exhibits sensitive dependence on initial conditions; that is, a small alteration of the state at one time will lead eventually to a state differing considerably from the state that would have occurred if the alteration had not been made. The atmosphere cannot be examined directly for chaos, but numerical studies with models with various degrees of complexity leave little doubt that the atmosphere is chaotic. Indeed, the nonlinearity introduced by the presence of advective processes---the only nonlinearity in some of the simpler models---is by itself quite sufficient to produce chaos.

The most obvious and most familiar consequence of atmospheric chaos is the limitation that it places on the possibility of forecasting most aspects of the weather pattern at long range, say two weeks or more in advance, in view of the impossibility of starting from a perfect analysis or using a perfect extrapolation procedure. However, there are more fundamental changes in dynamical thinking that the recognition of chaos has brought about.

For example, we now realize that many of the equations that we would like to solve---even some rather simples ones---do not possess general solutions that can be expressed in analytic form. If we have succeeded in analytically determining particular solutions, presumably steady or periodic ones, and if our equations are realistic enough to have captured the atmosphere's chaotic nature in their general solutions, our solutions will be highly specialized ones, and their relevance to the real atmosphere will at least be suspect.

Should we conclude, for example, at least in the context of a model that we are using, that the long-term average transport of angular momentum across middle latitudes is poleward, if a particular solution that we have found analytically should show a poleward transport? The real atmosphere does not exhibit a continual poleward transport, but instead possesses periods of poleward and also of equatorward transport, with the former dominating, and an extended solution of any realistic model should be expected to behave likewise. One property of chaotic solutions is that if one can identify a segment, and there should be many such segments, where the initial and final states are very much alike, a slight alteration of the initial state will produce a segment where they are exactly alike, i.e., a periodic solution, albeit an unstable one. Hence, we might have happened upon an analytic solution in which equatorward transport prevailed, rather than the solution that we did find. This being the case, can we show that the solutions with poleward transport are in some sense more representative? Are they, although unstable, perhaps less unstable? Here are plenty of problems left for the dynamicist to think about. It also appears that, more generally, we must meticulously avoid obtaining several numerical solutions for any problem, and then concluding without further inquiry that the ones that support a previously conceived hypothesis are the more representative ones. Certainly we must avoid acknowledging only these solutions in our write-up, even though we could do so without falsifying any results. Similarly, if we have been forced to base our conclusions on a single chaotic solution, we should be aware that another chaotic solution of the same equations might have led us to conclude something else.

So far I have been stressing the possibility that a system once thought to be behaving regularly may be found to be chaotic. We should not ignore the possibility that some systems once thought to be random may be found to be chaotic, in which case we may be able to discover their underlying dynamics. Chaos indeed has its positive side as well as its negative one…. Already more attention is being paid to the stratosphere and the tropical troposphere, where typical processes seem to be less nonlinear, and to mesoscale and smaller-scale systems, where they may be even more nonlinear. As for longer-period behavior, we do not yet know to what extent the progressive changes of climate, other than those associated with changing external conditions, are determined by the climate itself, and to what extent they are mere statistical residuals of shorter-period fluctuations. I feel confident that dynamic meteorology will provide the answer in the coming years (Lorenz, 1996).

The widespread discussion today of thresholds or “tipping points” is a characteristic of chaotic nonlinear systems, in mathematical systems, in sudden and potentially large changes in climate regimes, and perhaps too in attitudes and collective behavior.


Discussion Questions


a. Why is the concept of “deterministic nonperiodic flow” important in weather and climate studies?


b. What is “chaos theory” and where does it apply both within and beyond climate science?


c. How can simple mathematical models reveal basic truths not necessarily addressed in more complex models?


Reference

Lorenz, E.N., 1993 The Essence of Chaos. Seattle, University of Washington Press.

Lorenz, E.N., 1996. The evolution of dynamic meteorology. Historical Essays on Meteorology, 1919-1995, J.R. Fleming, ed. Boston, American Meteorological Society, 3-19.




Select articles citing this paper

Alexandru, A., R. de Elia, et al. (2007). "Internal variability in regional climate downscaling at the seasonal scale." Monthly Weather Review 135(9): 3221-3238.

Chaudhuri, S. (2007). "Chaotic graph theory approach for identification of convective available potential energy (CAPe) patterns required for the genesis of severe thunderstorms." Advances in Complex Systems 10(3): 413-422.

Cross, M. C. and P. C. Hohenberg (1993). "PATTERN-FORMATION OUTSIDE OF EQUILIBRIUM." Reviews of Modern Physics 65(3): 851-1112.

Eckmann, J. P. and D. Ruelle (1985). "ERGODIC-THEORY OF CHAOS AND STRANGE ATTRACTORS." Reviews of Modern Physics 57(3): 617-656.

Grassberger, P. and I. Procaccia (1983). "MEASURING THE STRANGENESS OF STRANGE ATTRACTORS." Physica D 9(1-2): 189-208.

Grassberger, P. and I. Procaccia (1983). "CHARACTERIZATION OF STRANGE ATTRACTORS." Physical Review Letters 50(5): 346-349.

Hohenegger, C. and C. Schar (2007). "Atmospheric predictability at synoptic versus cloud-resolving scales." Bulletin of the American Meteorological Society 88(11): 1783-+.

Ivanov, L. M. and P. C. Chu (2007). "On stochastic stability of regional ocean models with uncertainty in wind forcing." Nonlinear Processes in Geophysics 14(5): 655-670.

May, R. M. (1976). "SIMPLE MATHEMATICAL-MODELS WITH VERY COMPLICATED DYNAMICS." Nature 261(5560): 459-467.

Wolf, A., J. B. Swift, et al. (1985). "DETERMINING LYAPUNOV EXPONENTS FROM A TIME-SERIES." Physica D 16(3): 285-317.


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