Atmospheric Predictability: Why Butterflies Are Not of Practical Importance

The spectral turbulence model of Lorenz, as modified for surface quasigeostrophic dynamics by Rotunno and Snyder, is further modified to more smoothly approach nonlinear saturation. This model is used to investigate error growth starting from different distributions of the initial error. Consistent with an often overlooked finding by Lorenz, the loss of predictability generated by initial errors of small but fixed absolute magnitude is essentially independent of their spatial scale when the background saturation kinetic energy spectrum is proportional to the −5/3 power of the wavenumber. Thus, because the background kinetic energy increases with scale, very small relative errors at long wavelengths have similar impacts on perturbation error growth as large relative errors at short wavelengths. To the extent that this model applies to practical meteorological forecasts, the influence of initial perturbations generated by butterflies would be swamped by unavoidable tiny relative errors in the large scales. The rough applicability of the authors’ modified spectral turbulence model to the atmosphere over scales ranging between 10 and 1000 km is supported by the good estimate that it provides for the ensemble error growth in state-of-the-art ensemble mesoscale model simulations of two winter storms. The initial-error spectrum for the ensemble perturbations in these cases has maximum power at the longest wavelengths. The dominance of large-scale errors in the ensemble suggests that mesoscale weather forecasts may often be limited by errors arising from the large scales instead of being produced solely through an upscale cascade from the smallest scales.