Uncertainty and predictability in geophysics: Chaos and multifractal insights

Uncertainty and error growth are crosscutting geophysical issues. Since the "chaos revolution" the dominant paradigm has been the "butterfly effect": the dependence on initial conditions is so sensitive that errors grow exponentially fast with characteristic times. This was the outcome of studying superficially simple caricatures of more involved systems. We critically analyze the physical relevance of these models and the mathematical generality of this effect. We emphasize that the atmosphere, oceans, rain etc., are spatially extended turbulent systems, with wide ranges of spatial scales. Turbulent phenomenology already shows that errors grow only slowly across these scales; they follow power laws, there are no characteristic times. An important recent realization is that in spite of strong anisotropies the dynamically significant range of scales is much larger than previously thought and that the role of intermittency is drastic and yields much more frequent extremes. The focus is now on time-space geophysical scaling behavior: their multifractality. It is found quite generally - not only for turbulent fields- that an infinite hierarchy of exponents is required to characterize the predictability decay from average to extreme events. Nevertheless, these laws are meaningful over the whole time range from short to long term; we give their explicit expression. This multifractal predictability suggests the advantages of stochastic rather than deterministic sub-grid parametrizations, and makes stochastic forecasting very attractive.