The real butterfly effect

Historical evidence is reviewed to show that what Ed Lorenz meant by the iconic phrase 'the butterfly effect' is not at all captured by the notion of sensitive dependence on initial conditions in low-order chaos. Rather, as presented in his 1969 Tellus paper, Lorenz intended the phrase to describe the existence of an absolute finite-time predicability barrier in certain multi-scale fluid systems, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times. To distinguish from 'mere' sensitive dependence, the effect discussed in Lorenz's Tellus paper is referred to as 'the real butterfly effect'. Theoretical evidence for such a predictability barrier in a fluid described by the three-dimensional Navier–Stokes equations is discussed. Whilst it is still an open question whether the Navier–Stokes equation has this property, evidence from both idealized atmospheric simulators and analysis of operational weather forecasts suggests that the real butterfly effect exists in an asymptotic sense, i.e. for initial-time atmospheric perturbations that are small in scale and amplitude compared with (weather) scales of interest, but still large in scale and amplitude compared with variability in the viscous subrange. Despite this, the real butterfly effect is an intermittent phenomenon in the atmosphere, and its presence can be signalled a priori, and hence mitigated, by ensemble forecast methods.