Can a butterfly flutter in Rome start a tornado in Texas ? This is a version of the highly popularised statement about chaoticity of the atmosphere and the impossibility to have long term weather predictions.
Obviously, wikipedia clarifies, the flutter is the trigger of the phenomenon, but the evolution depends on the forces (energies) in act.
But is it really true ?
Actually to prove it one could simply take a modern weather model and put such variation in its initial conditions to see how the weather evolve in time and if such a perturbation can drive large variations in predictions.
In literature, in fact, this experiment seems not to exists, and the claim is based just on the properties of the Lorenz equations, which, indeed, are not a fully weather modeling but just one simplification. According to a livescience report, David Orrel, did this experiment, and found that the flutter of a butterfly is dissipated: so no butterfly effect. But I was not able to support it with the reading of Orrel's paper in bibliography. Maybe this is contained in his book, which I could not access.
Maybe such a small effect cannot be described within the allowed initial conditions of a weather model, and the question would be, in the case, meaningless. I am too ignorant in the field. But I suspect that this is the case.
Atmosphere is known to be chaotic, but the way it is, is described by the equations that the weather models contain, and how they can be initialised.
If the reality would be like described in the livescience post, and we believe in Paul Roebber, then we would driven to the conclusion that effects large enough to produce variations in the global weather system are as small as a single cloud (but not smaller). Therefore the atmosphere would be a very strange system (and maybe it is) where small perturbations (like the butterfly movements) are dissipated but larger perturbations (like uncertainty in the position of a cloud) grows chaotically.
I find strange that this issue was not investigated deeply, since, at least theoretically has some consequences.
In atmospheric sciences literature, chaos is studied by making an “ensemble” of simulations each one differing from the other by slightly different initial conditions, and some example are reported below in references.
P.S. Going more fundamental, part of the story would be to get to know if chaos exists in Navier-Stokes equations, and how sensitive it is. But this is for another post.
Alexanderian, A., Winokur, J., Sraj, I., Srinivasan, A., Iskandarani, M., Thacker, W. C., & Knio, O. M. (2012). Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach. Computational Geosciences, 16(3), 757–778. http://doi.org/10.1007/s10596-012-9286-2
Buizza, R. (2002). Chaos and weather prediction. Meteorological Training Course Lecture Series (pp. 1–28).
Orrel, D., Smith, L., Barkmeijer, J., & Palmer, T. N. (2002). Model error in weather forecasting Nonlinear Processes in Geophysics, 8, 357–371.
Privè, N. C., & Errico, R. M. (2013). The role of model and initial condition error in numerical weather forecasting investigated with an observing system simulation experiment. Tellus, 1–18. http://doi.org/10.3402/tellusa.v65i0.21740
Roebber, P. J., & Bosart, L. F. (1998). The Sensitivity of Precipitation to Circulation Details. Part I: An analysis of Regional Analogs. Monthly Weather Review, 126, 437–455.
Roebber, P. J., & Reuter, G. W. (2001). The sensitivity of Precipitation to Circulation Details. Part II: Mesoscale modeling. Monthly Weather Review, 130, 3–23.
Teixeira, J., Reynolds, C. A., & Judd, K. (2007). Time Step sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design. Journal of Atmospheric Sciences, 74, 175–191. http://doi.org/10.1175/JAS3824.1
Zhu, H., & Thorpe, A. (2006). Predictability of Extratropical Cyclones. The Influence of Initial Condition and model Uncertainties. Journal of the Atmospheric Sciences, 63, 1483–1497.