Revisiting Lorenz’s and Lilly’s Empirical Formulas for Predictability Estimates

Abstract. Recent studies have reiterated that the two-week predictability limit was originally estimated using a doubling time of five days from the Mintz-Arakawa model in the 1960s. However, this two-week predictability limit has conventionally been viewed as one of Lorenz's major findings from his 1969 studies. The limit has been presumably attributed to the mechanism involving the insignificant contributions of unresolved scales smaller than 38 meters. To understand the discrepancies in the origin of the two-week limit and to validate the mechanism in addressing the dependence of finite predictability on the atmospheric spectrum, we revisit Lorenz's studies, Lilly's work, and related research from the 1960s and early 1970s.

We first review how Lilly applied turnover time in turbulence theory to construct a convergent series that appears mathematically similar to the original Lorenz series. We then reexamine how Lorenz observed regularity in a sequence of saturation times over 21 selected wave modes and used the regularity to construct a convergent series, illustrating the negligible contribution of unresolved small-scale processes to predictability enhancement.

Our reanalysis does not support the claim that Lorenz’s and Lilly’s formulas are mathematically identical or physically comparable. Major discrepancies and inconsistencies include the use of different physical time scales in Lorenz's and Lilly's studies and the lack of a common factor of 2^{-2/3} that can be robustly determined from Lorenz's data. This falsifies the assumption that saturation time difference and turnover time are linearly proportional over the selected wave modes. Additionally, given the -5/3 power spectrum, we demonstrate that the convergence properties of Lorenz's or Lilly's series depend on spectral discretization. These issues, along with the highly simplified features of the Lorenz 1969 model, indicate that an upper bound for the predictability limit has not been robustly determined in Lorenz's and Lilly's studies. Therefore, caution should be exercised when applying Lilly's formula to conclude the dependence of finite predictability on the slopes of spectra. This perspective suggests opportunities to explore larger predictability and extend weather forecasts using various approaches, including sophisticated theoretical, real-world, and artificial intelligence-powered models.