Initial Error Growth and Predictability of Chaotic Low-dimensional Atmospheric Model

The growth of small errors in weather prediction is exponential on average. As an error becomes larger, its growth slows down and then stops with the magnitude of the error saturating at about the average distance between two states chosen randomly. This paper studies the error growth in a low-dimensional atmospheric model before, during and after the initial exponential divergence occurs. We test cubic, quartic and logarithmic hypotheses by ensemble prediction method. Furthermore, the quadratic hypothesis suggested by Lorenz in 1969 is compared with the ensemble prediction method. The study shows that a small error growth is best modeled by the quadratic hypothesis. After the error exceeds about a half of the average value of variables, logarithmic approximation becomes superior. It is also shown that the time length of the exponential growth in the model data is a function of the size of small initial error and the largest Lyapunov exponent. We conclude that the size of the error at the least upper bound (supremum) of time length is equal to 1 and it is invariant to these variables. Predictability, as a time interval, where the model error is growing, is for small initial error, the sum of the least upper bound of time interval of exponential growth and predictability for the size of initial error equal to 1.