The Real Butterfly Effect

There has been a renewed discussion of the relevance of the “butterfly effect” to describe the actual effect of the flapping of a butterfly wing on large-scale weather (on Real Climate see and on Climate Science see and see).

There is an important research issue with respect to the size of a perturbation of the atmosphere that must occur before it can have any effect on the larger-scale atmosphere. Ray Pierrehumbert and Gavin Schmidt on Real Climate conclude that there is no minimum spatial scale, while Issac Held states that features must be larger than a few millimeters.

Rich Eykholt and I have agreed to complete a paper on this subject over the coming months, as it clearly is an issue that has been neglected, and, in my view, is a misinterpretation of the conclusions from the seminal work of Ed Lorenz.

The real butterfly is illustrated below  

[from http://en.wikipedia.org/wiki/Chaos_theory]

“The Lorenz attractor is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.  Picture below is a plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3.”

from http://en.wikipedia.org/wiki/Chaos_theory

The interested reader can also evaluate the solution for different input values at  http://crossgroup.caltech.edu/chaos_new/Lorenz.html

In terms of what Professor Lorenz wrote, following is the text from his book The Essence of Chaos by Ed Lorenz in 1993 (from pages 14 and 15) regarding the expression “The Butterfly Effect”. The Figure 2 that he refers to in the text is of the form of the above two figures, and he labels it as “The butterfly”! Professor Lorenz wrote

“The expression has a somewhat cloudy history. It appears to have arisen following a paper that I presented at a meeting in Washington in 1972 entitled “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”  I avoided answering the question, but noted that if a single flap could lead to a tornado that would not otherwise have formed, it could equally well prevent a tornado that would otherwise have formed. I noted also that a single flap would have no more effect on the weather than any flap of any other butterfly’s wings, not to mention the activities of other species, including our own.  The paper is reproduced in its original form as Appendix A.

The thing that has made the origin of the phrase a bit uncertain is a peculiarity of the first chaotic system that I studied in detail.  Here an abbreviated graphical representation of a special collection of states known as a “strange attractor” was subsequently found to resemble a butterfly, and soon came to be known as the butterfly.   In Figure 2 we see one butterfly; a representative of a closely related species appears on the inside cover of Gleick’s book.  A number of people with whom I have talked have assumed that the butterfly effect was named after this attractor.  Perhaps it was.

Some correspondents have also called my attention to Ray Bradbury’s intriguing short story, “A Sound of Thunder,” written long before the Washington meeting.  Here the death of a prehistoric butterfly, and its consequent failure to reproduce, change the outcome of a present-day presidential election.

Before the Washington meeting, I had sometimes used a sea gull as a symbol for sensitive dependence.  The switch to a butterfly was made by the session convenor, the meteorologist Philip Merilees, who was unable to check with me when he had to submit the program titles. Phil has recently assured me that he was not familiar with Bradbury’s story. Perhaps the butterfly, with its seeming frailty and lack of power, is a natural choice for a symbol of the small that can produce the great.

Other symbols have preceded the sea gull. In George W. Stewart’s novel Storm, a copy of which my sister gave me for Christmas when she first learned I was to become a meteorology student, a meteorologist recalls his professor’s remark that a man sneezing in China may set people to shoveling snow in New York.  Stewart’s professor was simply echoing what some real-world meteorologists had been saying for many years, sometimes facetiously, sometimes seriously.”

Thus, the butterfly effect, which is described by the solution shape in the above figures, has morphed into a symbol that small perturbations can alter large-scale structure.

However, scientists such as Ray Pierrehumbert and Gavin Schmidt at Real Climate have literally interpreted Professor Lorenz’s seminal as applying to all perturbations of atmospheric flow regardless of their magnitude and spatial scale.  This clearly was not the claim of Professor Lorenz.

In the real world, very small perturbations, such as the flap of a butterfly wing cannot have any impact on the large-scale flow (such as the creation of a tornado). In order to do that, the turbulence generated by the flapping wings must retain some coherant flow structure as the nonlinear interactions create larger scale structure. However, this kinetic energy is dispersed over progressively larger and larger volumes such that it will quickly dissipate into heat as the magnitude of the disturbance to the flow at any single location becomes smaller. The atmosphere has an infinitesimal addition of heat, but the coherent information needed to alter the large-scale flow is lost.

This paragraph should, of course, be viewed as a hypothesis, and we will be evaluating this in our paper. Readers, including those at Real Climate, are invited to also seek to falsify this hypothesis.